1
Introduction to Stochastic Calculus
B V Rao and R L Karandikar
Chennai Mathematical Institute
[email protected] and [email protected]
About this book:
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Version dated 30 May 2017.
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The book could be described as Stochastic Integration without tears or fear or even as
Stochastic Integration made easy.
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This book can be used as a 2 semester graduate level course on Stochastic Calculus.
The background required is a course on measure theoretic probability.
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The book begins with conditional expectation and martingales and basic results on
martingales are included with proofs (in discrete time as well as in continuous time). Then
a chapter on Brownian motion and Ito integration w.r.t. Brownian motion follows, which
includes stochastic di↵erential equations. These three chapters over 80 pages give a soft
landing to a reader to the more complex results that follow. The first three chapters form
the Introductory material.
Taking a cue from the Ito integral, a Stochastic Integrator is defined and its properties
as well as the properties of the integral are defined at one go. In most treatments, we
start by defining the integral for a square integrable martingales and where integrands
themselves are in suitable Hilbert space. Then over several stages the integral is extended,
and at each step one has to reaffirm its properties. Here this is avoided. Just using the
definition, various results including quadratic variation, Ito formula, Emery topology are
defined and studied.
We then define quadratic variation of a square integrable martingale and using this show
that these are stochastic integrators. Using an inequality by Burkholder, we show that all
martingales and local martingales are stochastic integrators and thus semimartingales are
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stochastic integrators. We then show that stochastic integrators are semimartingales and
obtain various results such as a description of the class of integrands for the stochastic
integral. We complete the chapter by giving a proof of the Bichteler-Dellacherie-MeyerMokobodzky Theorem.
These two chapters (of about 100 pages) form the basic material. Most results used in
these two chapter are known, but we have put them in a order to make the presentation
easy flowing. We have avoided invoking results from functional analysis but rather included
the required steps. Thus instead of saying that the integral is a continuous linear functional
on a dense subset of a Banach space and hence can be extended to the Banach space, we
explicitly construct the extension!
Next we introduce Pathwise formulae for the stochastic integral. These as yet have not
found a place in any textbook on stochastic integration. We briefly specialise to continuous
semimartingales and obtain growth estimates and study solution of a stochastic di↵erential
equation (SDE) using the technique of random time change. We also prove a pathwise
formulae for the solution of an SDE driven by continuous semimartingales.
Then we move onto a study of predictable increasing process and introduce predictable
stopping times etc and prove the Doob Meyer decomposition theorem
We conclude by proving the Metivier Pellaumail inequality and using it introduce a
notion of dominating process of a semimartingale. We then obtain existence, uniqueness of
solutions to an SDE driven by a general semimartingale and also obtain a pathwise formula
by showing that modified Eular-Peano approximations converge almost surely.
Contents
1 Discrete Parameter Martingales
7
Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.2
Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.3
Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.4
Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.5
Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.6
Stop Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.7
Doob’s Maximal Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.8
Martingale Convergence Theorem . . . . . . . . . . . . . . . . . . . . . . . .
22
1.9
Square Integrable Martingales . . . . . . . . . . . . . . . . . . . . . . . . . .
29
1.10 Burkholder-Davis-Gundy inequality . . . . . . . . . . . . . . . . . . . . . . .
34
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1.1
40
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2 Continuous Time Processes
Notations and Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
2.2
Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
2.3
Martingales and Stop Times . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.4
A Version of Monotone Class Theorem . . . . . . . . . . . . . . . . . . . . .
59
2.5
The UCP Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
2.6
The Lebesgue-Stieltjes Integral . . . . . . . . . . . . . . . . . . . . . . . . .
66
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2.1
3 The Ito Integral
69
3.1
Quadratic Variation of Brownian Motion . . . . . . . . . . . . . . . . . . . .
69
3.2
Levy’s Characterization of the Brownian motion . . . . . . . . . . . . . . .
71
3.3
The Ito Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
3.4
Multidimensional Ito Integral . . . . . . . . . . . . . . . . . . . . . . . . . .
81
3.5
Stochastic Di↵erential Equations . . . . . . . . . . . . . . . . . . . . . . . .
85
3
4
CONTENTS
4 Stochastic Integration
91
4.1
The Predictable -Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
4.2
Stochastic Integrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
4.3
Properties of the Stochastic Integral . . . . . . . . . . . . . . . . . . . . . . 101
4.4
Locally Bounded Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.5
Approximation by Riemann Sums
4.6
Quadratic Variation of Stochastic Integrators . . . . . . . . . . . . . . . . . 119
4.7
Quadratic Variation of the Stochastic Integral . . . . . . . . . . . . . . . . . 126
4.8
Ito’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.9
The Emery Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
. . . . . . . . . . . . . . . . . . . . . . . 115
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4.10 Extension Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5 Semimartingales
159
Notations and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.2
The Quadratic Variation Map . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.3
Quadratic Variation of a Square Integrable Martingale . . . . . . . . . . . . 165
5.4
Square Integrable Martingales are Stochastic Integrators . . . . . . . . . . . 173
5.5
Semimartingales are Stochastic Integrators . . . . . . . . . . . . . . . . . . . 177
5.6
Stochastic Integrators are Semimartingales . . . . . . . . . . . . . . . . . . . 180
5.7
The Class L(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.8
The Bichteler-Dellacherie-Meyer-Mokobodzky Theorem . . . . . . . . . . . 198
5.9
Enlargement of Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
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5.1
210
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6 Pathwise Formula for the Stochastic Integral
6.1
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6.2
Pathwise Formula for the Stochastic Integral . . . . . . . . . . . . . . . . . 211
6.3
Pathwise Formula for Quadratic Variation . . . . . . . . . . . . . . . . . . . 214
7 Continuous Semimartingales
216
7.1
Random Time Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
7.2
Growth estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
7.3
Stochastic Di↵erential Equations . . . . . . . . . . . . . . . . . . . . . . . . 224
7.4
Pathwise Formula for solution of SDE . . . . . . . . . . . . . . . . . . . . . 236
7.5
Matrix-valued Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . . 238
CONTENTS
5
8 Predictable Increasing Processes
8.1
8.2
The -field F⌧
242
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
Predictable Stop Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8.3
Natural FV Processes are Predictable . . . . . . . . . . . . . . . . . . . . . 258
8.4
Decomposition of Semimartingales revisited . . . . . . . . . . . . . . . . . . 265
8.5
Doob-Meyer Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
8.6
Square Integrable Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9 The Davis Inequality
291
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
9.2
Burkholder-Davis-Gundy inequality - Continuous Time . . . . . . . . . . . . 293
9.3
On Stochastic Integral w.r.t. a Martingale . . . . . . . . . . . . . . . . . . . 299
9.4
Sigma-Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
9.5
Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
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9.1
10 Integral Representation of Martingales
308
10.3 Quasi-elliptical multidimensional Semimartingales
. . . . . . . . . . . . . . 317
10.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
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10.2 One Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
10.4 Continuous Multidimensional Semimartingales . . . . . . . . . . . . . . . . 322
10.5 General Multidimensional case . . . . . . . . . . . . . . . . . . . . . . . . . 326
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10.6 Integral Representation w.r.t. Sigma-Martingales . . . . . . . . . . . . . . . 339
10.7 Connections to Mathematical Finance . . . . . . . . . . . . . . . . . . . . . 342
347
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11 Dominating Process of a Semimartingale
11.1 An optimization result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
11.2 Metivier-Pellaumail inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 351
11.3 Growth Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
11.4 Alternate metric for Emery topology . . . . . . . . . . . . . . . . . . . . . . 359
12 SDE driven by r.c.l.l. Semimartingales
367
12.1 Gronwall Type Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
12.2 Stochastic Di↵erential Equations . . . . . . . . . . . . . . . . . . . . . . . . 370
12.3 Pathwise Formula for solution to an SDE . . . . . . . . . . . . . . . . . . . 377
12.4 Euler-Peano approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
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CONTENTS
12.5 Matrix-valued Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . . 389
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13 Girsanov Theorem
393
13.1 Girsanov Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
Chapter 1
Discrete Parameter Martingales
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In this chapter, we will discuss martingales indexed by integers (mostly positive integers)
and obtain basic inequalities on martingales and other results which are the basis of most of
the developments in later chapters on stochastic integration. We will begin with a discussion on conditional expectation and then on filtration two notions central to martingales.
Notations
A
1.1
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For an integer d 1, Rd denotes the d-dimensional Euclidean space and B(Rd ) will denote
the Borel -field on Rd . Further, C(Rd ) and Cb (Rd ) will denote the classes of continuous
functions and bounded continuous functions on Rd respectively. When d = 1, we will write
R in place of R1 .
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(⌦, F, P) will denote a generic probability space and B(⌦, F) will denote the class of
real-valued bounded F-measurable functions.
For a collection A ✓ F, (A) will denote the smallest -field which contains A and
for a collection G ✓ B(⌦, F), (G) will likewise denote the smallest -field with respect to
which each function in G is measurable.
It is well known and easy to prove that
(Cb (Rd )) = B(Rd ).
An Rd -valued random variable X (on a probability space (⌦, F, P)) is a measurable
function from (⌦, F) to (Rd , B(Rd )). For such an X and a function f 2 Cb (Rd ), E[f (X)]
(or EP [f (X)] if there are more than one probability measure under consideration) will
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Chapter 1. Discrete Parameter Martingales
denote the integral
E[f (X)] =
Z
f (X(!))dP(!).
⌦
For any measure µ on (⌦, A) and for 1  p < 1, we will denote by Lp (µ) the space
Lp (⌦, A, µ) of real-valued A measurable functions equipped with the norm
Z
1
kf kp = ( |f |p dµ) p .
It is well known that Lp (µ) is a Banach space under the norm kf kp .
For more details and discussions as well as proofs of statements quoted in this chapter,
see Billingsley [4], Breiman [5], Ethier and Kurtz[18].
Conditional Expectation
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1.2
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Let X and Y be random variables. Suppose we are going to observe Y and are required to
make a guess for the value of X. Of course, we would like to be as close to X as possible.
Suppose the penalty function is square of the error. Thus we wish to minimize
E[(X
a)2 ]
(1.2.1)
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A
where a is the guess or the estimate. For this to be meaningful, we should assume E[X 2 ] <
1. The value of a that minimizes (1.2.1) is the mean µ = E[X]. On the other hand, if we
are allowed to use observations Y while making the guess, then our estimate could be a
function of Y . Thus we should choose the function g such that
E[(X
g(Y ))2 ]
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takes the minimum possible value. It can be shown that there exists a function g (Borel
measurable function from R to R) such that
E[(X
g(Y ))2 ]  E[(X
f (Y ))2 ]
(1.2.2)
for all (Borel measurable) functions f . Further, if g1 , g2 are two functions satisfying (1.2.2),
then g1 (Y ) = g2 (Y ) almost surely P. Indeed, A = L2 (⌦, F, P) - the space of all square
integrable random variables on (⌦, F, P) with inner product hX, Y i = E[XY ] giving rise
p
to the norm kZk = E[Z 2 ] is a Hilbert space and
K = {f (Y ) : f from R to R measurable, E[(f (Y ))2 ] < 1}
is a closed subspace of A. Hence given X 2 A, there is a unique element in K, namely
the orthogonal projection of X to K, that satisfies (1.2.2). Thus for X 2 A, we can define
1.2. Conditional Expectation
9
g(Y ) to be the conditional expectation of X given Y , written as E[X | Y ] = g(Y ). One
can show that for X, Z 2 A and a, b 2 R
E[aX + bZ | Y ] = aE[X | Y ] + bE[Z | Y ]
and
X  Z implies E[X | Y ]  E[Z | Y ].
Note that (1.2.2) implies that for all t
E[(X
g(Y ))2 ]  E[(X
g(Y ) + tf (Y ))2 ]
for any f (Y ) 2 K and hence that
t2 E[f (Y )2 ] + 2tE[(X
g(Y ))f (Y )]
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In particular, g(Y ) satisfies
0, 8t.
g(Y ))f (Y )] = 0 8f bounded measurable.
E[(X
(1.2.3)
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Indeed, (1.2.3) characterizes g(Y ). Also, (1.2.3) is meaningful even when X is not square
integrable but only E[|X| ] < 1. With a little work we can show that given X with
E[|X| ] < 1, there exists a unique g(Y ) such that (1.2.3) is valid. To see this, suffices to
consider X
0, and then taking Xn = min(X, n), gn such that E[Xn | Y ] = gn (Y ), it
follows that gn (Y )  gn+1 (Y ) a.s. and hence defining g̃(x) = lim sup gn (x), g(x) = g̃(x) if
g̃(x) < 1 and g(x) = 0 otherwise, one can show that
g(Y ))f (Y )] = 0 8f bounded measurable.
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E[(X
D
It is easy to see that in (1.2.3) it suffices to take f to be {0, 1}-valued, i.e. indicator function
of a Borel set. We are thus led to the following definition: For random variables X, Y with
E[|X| ] < 1,
E[X | Y ] = g(Y )
where g is a Borel function satisfying
E[(X
g(Y ))1B (Y )] = 0, 8B 2 B(R).
(1.2.4)
Now if instead of one random variable Y , we were to observe Y1 , . . . , Ym , we can similarly
define
E[X | Y1 , . . . Ym ] = g(Y1 , . . . Ym )
where g satisfies
E[(X
g(Y1 , . . . Ym ))1B (Y1 , . . . Ym )] = 0, 8B 2 B(Rm ).
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Chapter 1. Discrete Parameter Martingales
Also if we were to observe an infinite sequence, we have to proceed similarly, with g being
a Borel function on R1 . Of course, the random variables could be taking values in Rd .
In each case we will have to write down properties and proofs thereof and keep doing the
same as the class of observable random variables changes.
Instead, here is a unified way. Let (⌦, F, P) be a probability space and Y be a random
variable on ⌦. The smallest -field (Y ) with respect to which Y is measurable (also called
the -field generated by Y ) is given by
(Y ) = {A 2 F : A = {Y 2 B}, B 2 B(R)}.
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A random variable Z can be written as Z = g(Y ) for a measurable function g if and only
if Z is measurable with respect to (Y ). Likewise, for random variables Y1 , . . . Ym , the
-field (Y1 , . . . Ym ) generated by Y1 , . . . Ym is the smallest -field with respect to which
Y1 , . . . Ym are measurable and is given by
(Y1 , . . . Ym ) = {A 2 F : A = {(Y1 , . . . Ym ) 2 B}, B 2 B(Rm )}.
A
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Similar statement is true even for an infinite sequence of random variables. In view of
these observations, one can define conditional expectation given a -field as follows. It
should be remembered that one mostly uses it when the -field in question is generated by
a collection of observable random variables.
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Definition 1.1. Let X be a random variable on (⌦, F, P) with E[|X| ] < 1 and let G be
a sub- -field of F. Then the conditional expectation of X given G is defined to be the G
measurable random variable Z such that
E[X 1A ] = E[Z 1A ], 8A 2 G.
(1.2.5)
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Existence of Z can be proven on the same lines as given above, first for the case when
X is square integrable and then for general X. Also, Z is uniquely determined up to null
sets if Z and Z 0 are G measurable and both satisfy (1.2.5), then P(Z = Z 0 ) = 1. Some
properties of conditional expectation are given in the following two propositions.
Throughout the book, we adopt a convention that all statements involving random variables are to be interpreted in almost sure sense- i.e. X = Y means X = Y a.s., X  Y
means X  Y a.s. and so on.
Proposition 1.2. Let X, Z be integrable random variables on (⌦, F, P) and G, H be sub-fields of F with G ✓ H and a, b 2 R. Then we have
(i) E[aX + bZ | G] = aE[X | G] + bE[Z | G].
1.3. Independence
11
(ii) X  Z ) E[X | G]  E[Z | G].
(iii) |E[X | G]|  E[|X| | G].
(iv) E[E[X | G]] = E[X].
(v) E[E[X | H] | G] = E[X | G].
(vi) If Z is G measurable such that E[ |ZX| ] < 1 then
E[ZX | G] = ZE[X | G].
Of course when X is square integrable, we do have an analogue of (1.2.2):
E[(X
E[X | H])2 ]  E[(X
Independence
U )2 ].
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1.3
-field.
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Proposition 1.3. Let X be a random variable with E[X 2 ] < 1 and H be a sub
Then for all H measurable square integrable random variables U
A
Two events A, B in a probability space (⌦, F, P), i.e. A, B 2 F are said to be independent
if
P(A \ B) = P(A)P(B).
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For j = 1, 2, . . . m, let Xj be an Rd -valued random variable on a probability space (⌦, F, P).
Then X1 , X2 , . . . Xm are said to be independent if for all Aj 2 B(Rd ), 1  j  m
P(Xj 2 Aj , 1  j  m) =
m
Y
j=1
P(Xj 2 Aj ).
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A sequence {Xn } of random variables is said to be a sequence of independent random
variables if X1 , X2 , . . . , Xm are independent for every m 1.
Let G be a sub -field of F. An Rd -valued random variable X is said to be independent
of the - field G if for all A 2 B(Rd ), D 2 G,
P({X 2 A} \ D) = P({X 2 A})P(D).
Exercise 1.4. Let X, Y be real valued random variables. Show that
(i) X, Y are independent if and only if for all bounded Borel measurable functions f, g on R,
one has
E[f (X)g(Y )] = E[f (X)]E[g(Y )].
(1.3.1)
12
Chapter 1. Discrete Parameter Martingales
(ii) X, Y are independent if and only if for all bounded Borel measurable functions f, on R,
one has
E[f (X) | (Y )] = E[f (X)].
(1.3.2)
(iii) X, Y are independent if and only if for all t 2 R, one has
E[exp{itX} | (Y )] = E[exp{itX}].
(1.3.3)
Exercise 1.5. Let U be an Rd -valued random variable and let G be a -field. Show that U
is independent of G if and only if for all 2 Rd
E[exp{i · U } | G] = E[exp{i · U }].
Filtration
T
1.4
(1.3.4)
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Suppose Xn denotes a signal being transmitted at time n over a noisy channel (such as
voice over telephone lines) and let Nn denote the noise at time n and Yn denote the noise
corrupted signal that is observed. Under the assumption of additive noise, we get
0.
A
Yn = Xn + Nn , n
D
R
Now the interest typically is in estimating the signal Xn at time n. Since the noise as well
the true signal are not observed, we must require that the estimate X̂n of the signal at
time n be a function of only observations upto time n, i.e. X̂n must only be a function of
{Yk : 0  k  n}, or X̂n is measurable with respect to the -field Gn = {Yk : 0  k  n}.
A sequence of random variables X = {Xn } will also be referred to as a process. Usually, the
index n is interpreted as time. This is the framework for filtering theory. See Kallianpur
[30] for more on filtering theory.
Consider a situation from finance. Let Sn be the market price of shares of a company
UVW at time n. Let An denote the value of the assets of the company, Bn denote the value
of contracts that the company has bid, Cn denote the value of contracts that the company
is about to sign. The process S is observed by the public but the processes A, B, C are not
observed by the public at large. Hence, while making a decision on investing in shares of the
company UVW, on the nth day, an investor can only use information {Sk : 0  k < n} (we
assume that the investment decision is to be made before the price on nth day is revealed).
Indeed, in trying to find an optimal investment policy ⇡ = (⇡n ) (optimum under some
criterion), the class of all investment strategies must be taken as all processes ⇡ such that
1.4. Filtration
13
for each n, ⇡n is a (measurable) function of {Sk : 0  k < n}. In particular, the strategy
cannot be a function of the unobserved processes A, B, C.
Thus it is useful to define for n 0, Gn to be the -field generated by all the random
variables observable before time n namely, S0 , S1 , S2 , . . . , Sn 1 and then require any action
to be taken at time n (say investment decision) should be measurable with respect to Gn .
These observations lead to the following definitions
Definition 1.6. A filtration on a probability space (⌦, F, P) is an increasing family of sub
-fields (F⇧ ) = {Fn : n 0} of F indexed by n 2 {0, 1, 2, . . . , m, . . .}.
Definition 1.7. A stochastic process X i.e. a sequence X={Xn } of random variables is said
to be adapted to a filtration (F⇧ ) if for all n 0, Xn is Fn measurable.
F
T
We will assume that the underlying probability is complete i.e. N 2 F, P(N ) = 0 and
N1 ✓ N implies N1 2 F and that F0 contains all sets N 2 F with P(N ) = 0. We will
refer to a stochastic process as a process. Let N be the class of all null sets (sets with
P(N ) = 0), and for a process Z, possibly vector-valued, let
FnZ = (Zk : 0  k  n)
(1.4.1)
FenZ = (N [ FnZ ).
(1.4.2)
A
and
While it is not required in the definition, in most situations, the filtration (F⇧ ) under
consideration would be chosen to be (for a suitable process Z)
0}
(Fe⇧Z ) = {FenZ : n
0}.
R
D
or
(F⇧Z ) = {FnZ : n
Sometimes, a filtration is treated as a mere technicality. We would like to stress that
it is not so. It is a technical concept, but a very important ingredient of the analysis. For
example, in the estimation problem, one could consider the filtration (Fe⇧X,N ) (recall, X is
signal, N is noise and Y = X + N is the observable) as well as (F⇧Y ). While one can use
(Fe⇧X,N ) for technical reasons say in a proof, but when it comes to estimating the signal, the
estimate at time n has to be measurable with respect to FenY . If X̂n represents the estimate
of the signal at time n, then the process X̂ must be required to be adapted to (Fe⇧Y ).
Requiring it to be adapted to (Fe⇧X,N ) is not meaningful. Indeed, if we can take X̂n to be
adapted to (Fe⇧X,N ), then we can as well take X̂n = Xn which is of course not meaningful.
Thus, here (Fe⇧X,N ) is a mere technicality while (Fe⇧Y ) is more than a technicality.
14
1.5
Chapter 1. Discrete Parameter Martingales
Martingales
In this section, we will fix a probability space (⌦, F, P) and a filtration (F⇧ ). We will only
be considering R-valued processes in this section.
Definition 1.8. A sequence M = {Mn } of random variables is said to be a martingale if
M is (F⇧ ) adapted and for n 0 one has E[|Mn | ] < 1 and
EP [Mn+1 | Fn ] = Mn .
Definition 1.9. A sequence M = {Mn } of random variables is said to be a submartingale
if M is (F⇧ ) adapted and for n 0 one has E[|Mn | ] < 1 and
EP [Mn+1 | Fn ]
Mn .
F
T
When there are more than one filtration in consideration, we will call it a (F⇧ )-martingale
or martingale w.r.t. (F⇧ ). Alternatively, we will say that {(Mn , Fn ) : n 1} is a martingale.
It is easy to see that for a martingale M , for any m < n, one has
EP [Mn | Fm ] = Mm
D
R
A
and similar statement is also true for submartingales. Indeed, one can define martingales
and submartingales indexed by an arbitrary partially ordered set. We do not discuss these
in this book.
If M is a martingale and is a convex function on R, then Jensen’s inequality implies
that the process X = {Xn } defined by Xn = (Mn ) is a submartingale if Xn is integrable
for all n. If M is a submartingale and is an increasing convex function then X is also
a submartingale provided Xn is integrable for each n. In particular, if M is a martingale
or a positive submartingale with E[Mn2 ] < 1 for all n, then Y defined by Yn = Mn2 is a
submartingale.
When we are having only one filtration under consideration, we will drop reference to
it and simply say M is a martingale. It is easy to see also that sum of two martingales with
respect to the same underlying filtration is also a martingale. We note here an important
property of martingales that would be used later.
Theorem 1.10. Let M m be a sequence of martingales on some probability space (⌦, F, P)
w.r.t. a fixed filtration (F⇧ ). Suppose that
Mnm ! Mn in L1 (P), 8n
Then M is also a martingale w.r.t. the filtration (F⇧ ).
0.
1.5. Martingales
15
Proof. Note that for any X m converging to X in L1 (P), for any -field G, using (i), (ii),
(iii) and (iv) in Proposition 1.2, one has
E[|E[X m | G]
E[X | G]| ] = E[|E[(X m
 E[E[|X
X) | G]| ]
m
X| | G]]
= E[|X m
X| ]
! 0.
For n
0, applying this to X m = Mnm , one gets
m
Mnm = E[Mn+1
| Fn ] ! E[Mn+1 | Fn ] in L1 (P).
But Mnm ! Mn in L1 (P) so that E[Mn+1 | Fn ] = Mn . It follows that M is a martingale.
F
T
The following decomposition result, called the Doob decomposition, is simple to prove
but its analogue in continuous time was the beginning of the theory of stochastic integration.
A
Theorem 1.11. Let X be a submartingale. Let A = {An } be defined by A0 = 0 and for
n 1,
n
X
An =
E[(Xk Xk 1 ) | Fk 1 ].
k=1
R
Then A is an increasing process (i.e. An  An+1 for n
measurable for each n and M = {Mn } defined by
Mn = X n
0 ) such that A0 = 0, An is Fn
An
D
is a martingale. Further, if B = {Bn } is a process such that B0 = 0, Bn is Fn
for each n and N = {Nn } defined by Nn = Xn Bn is a martingale, then
P(An = Bn 8n
1
1
measurable
1) = 1.
Proof. Since X is a submartingale, each summand in the definition of An is non-negative
and hence A is an increasing process. Clearly, An is Fn 1 measurable. By the definition
of An , Mn we can see that
Mn
Mn
1
= Xn
Xn
E[Mn
Mn
1
E[Xn
Xn
and hence that
1 | Fn 1 ]
=0
1 | Fn 1 ]
16
Chapter 1. Discrete Parameter Martingales
showing that M is a martingale. If Bn is as in the statement we can see that for n
E[(Xn
Xn
1 ) | Fn 1 ]
= E[(Nn
Nn
1 ) | Fn 1 ]
= E[(Bn
Bn
1 ) | Fn 1 ]
= Bn
Now B0 = 0 implies
Bn =
Bn
n
X
Bn
1 ) | Fn 1 ]
1.
E[(Xk
Xk
1)
k=1
completing the proof.
+ E[(Bn
1,
| Fk
1]
1]
= 0 8n
1.
(1.5.1)
F
E[Dn | Fn
T
The uniqueness in the result above depends strongly on the assumption that Bn is Fn 1
measurable. The process A is called the compensator of the submartingale X.
Let M = {Mn } be a martingale. The sequence D defined by Dn = Mn Mn 1 , for
n 1 and D0 = M0 is called a martingale di↵erence sequence and satisfies
Definition 1.12. A sequence of random variables U = {Un } is said to be a predictable if for
all n 1, Un is Fn 1 measurable and U0 is F0 measurable.
D
R
A
The compensator A appearing in the Doob decomposition of a submartingale M is
predictable.
Consider a gambling house, where a sequence of games are being played, at say one
hour interval. If an amount a is bet on the nth game, the reward on the nth game is aDn .
Since a gambler can use the outcomes of the games that have been played earlier, Un
the amount she bets on nth game can be random instead of a fixed number but has to be
predictable with respect to the underlying filtration since the gambler has to decide how
much to bet before the nth game is played. If the game is fair, i.e. E[Dn | Fn 1 ] = 0, then
the partial sums Mn = D0 + . . . + Dn is a martingale and the total reward Rn at time n
P
is then given by Rn = nk=0 Uk Dk . One can see that it is also a martingale, if say Un is
bounded. This leads to the following definition.
Definition 1.13. Let M = {Mn } be a martingale and U = {Un } be a predictable sequence
of random variables. The process Z = {Zn } defined by Z0 = 0 and for n 1
Zn =
n
X
Uk (Mk
Mk
1)
k=1
is called the martingale transform of M by the sequence U .
(1.5.2)
1.5. Martingales
17
The following result gives conditions under which the transformed sequence is a martingale.
Theorem 1.14. Suppose M = {Mn } is a martingale and U = {Un } is a predictable
sequence of random variables such that
E[|Mn Un | ] < 1 for all n
1.
(1.5.3)
Then the martingale transform Z defined by (1.5.2) is a martingale.
Proof. Let us note that
E[|Mn Un | ] = E[E[|Mn Un | | Fn
1 ]]
1 ]| ]
= E[|Un E[Mn | Fn
1 ]| ]
= E[|Un Mn
T
E[|E[Mn Un | Fn
(1.5.4)
1| ]
E[Un (Mn
Mn
A
F
where we have used properties of conditional expectation and the fact that Un is Fn 1
measurable and that M is a martingale. Thus, (1.5.3) implies E[|Un Mn 1 | ] < 1. This is
needed to justify splitting the expression in the next step.
1 ) | Fn 1 ]
= E[Un Mn | Fn
1]
E[Un Mn
= Un E[Mn | Fn
1]
U n Mn
R
= U n Mn
1
U n Mn
1 | Fn 1 ]
1
(1.5.5)
1
= 0.
Mn
1)
is a martingale di↵erence sequence and thus Z defined
D
This implies Cn = Un (Mn
by (1.5.2) is a martingale.
The proof given above essentially also yields the following:
Theorem 1.15. Suppose M = {Mn } is a submartingale and U = {Un } is a predictable
sequence of random variables such that Un 0 and
E[|Mn Un | ] < 1 for all n
1.
Then the transform Z defined by (1.5.2) is a submartingale.
Exercise 1.16. Let (Mn , Fn ) be a martingale. Show that M is a (F·M ) martingale.
(1.5.6)
18
1.6
Chapter 1. Discrete Parameter Martingales
Stop Times
We continue to work with a fixed probability space (⌦, F, P) and a filtration (F⇧ ).
Definition 1.17. A stop time ⌧ is a function from ⌦ into {0, 1, 2, . . . , } [ {1} such that for
all n 0,
{⌧ = n} 2 Fn , 8n < 1.
T
Equivalently, ⌧ is a stop time if {⌧  n} 2 Fn for all n 1. stop times were introduced
in the context of Markov Chains by Doob. Martingales and stop times together are very
important tools in the theory of stochastic process in general and stochastic calculus in
particular.
F
Definition 1.18. Let ⌧ be a stop time and X be an adapted process. The stopped random
variable X⌧ is defined by
1
X
X⌧ (!) =
Xn (!)1{⌧ =n} .
n=0
A
Note that by definition, X⌧ = X⌧ 1{⌧ <1} . The following results connecting martingales
and submartingales and stop times (and their counterparts in continuous time) play a very
important role in the theory of stochastic processes.
and ⌧ be two stop times and let
R
Exercise 1.19. Let
⇠=⌧_
and ⌘ = ⌧ ^ .
D
Show that ⇠ and ⌘ are also stop times. Here and in the rest of this book, a _ b = max(a, b)
and a ^ b = min(a, b).
Exercise 1.20. Let ⌧ be a random variable taking values in {0, 1, 2, . . .} and for n 0 let
Fn = (⌧ ^ n). If ⌘ is a stop time for this filtration then show that ⌘ = (⌘ _ ⌧ ) ^ r for some
r 2 {0, 1, 2 . . . , 1}
Theorem 1.21. Let M = {Mn } be a submartingale and ⌧ be a stop time. Then the process
N = {Nn } defined by
Nn = Mn^⌧ ,
is a submartingale. Further, if M is a martingale then so is N .
1.6. Stop Times
19
Proof. Without loss of generality, we assume that M0 = 0. Let Un = 1{⌧ <n} and Vn =
1{⌧ n} . Since
{Un = 1} = [nk=01 {⌧ = k}
it follows that Un is Fn 1 measurable and hence U is a predictable sequence and since
Un + Vn = 1, it follows that V is also a predictable sequence. Noting that N0 = 0 and for
n 1
n
X
Nn =
Vk (Mk Mk 1 )
k=1
the result follows from Theorem 1.14.
The following version of this result is also useful.
Mn^⌧ ,
 ⌧.
F
S n = Mn
T
Theorem 1.22. Let M = {Mn } be a submartingale and , ⌧ be stop times such that
Let R = {Rn }, S = {Sn } be defined as follows: R0 = S0 = 0 and for n 1
Rn = Mn^⌧
Mn^ .
Then R, S are submartingales. Further, if M is a martingale then so are S, R.
A
Proof. To proceed as earlier, let Un = 1{⌧ <n} , Vn = 1{ <n⌧ } = 1{⌧
again U, V are predictable and note that S0 = R0 = 0 and for n 1
R
Sn =
D
Rn =
n
X
k=1
n
X
Uk (Mk
Mk
1)
Vk (Mk
Mk
1 ).
n}
1{
n} .
Once
k=1
The result follows from Theorems 1.14.
The N in Theorem 1.21 is the submartingale M stopped at ⌧ . S in Theorem 1.22 is
the increment of M after ⌧ .
Corollary 1.23. Let M = {Mn } be a submartingale and , ⌧ be stop times such that
 ⌧ . Then for all n 1
E[Mn^ ]  E[Mn^⌧ ].
It is easy to see that for a martingale M , E[Mn ] = E[M0 ] for all n 1. Of course, this
property does not characterize martingales. However, we do have the following result.
20
Chapter 1. Discrete Parameter Martingales
Theorem 1.24. Let M = {Mn } be an adapted process such that E[|Mn | ] < 1 for all
n 0. Then M is a martingale if and only if for all bounded stop times ⌧ ,
E[M⌧ ] = E[M0 ].
(1.6.1)
Proof. If M is a martingale, Theorem 1.21 implies that
E[M⌧ ^n ] = E[M0 ].
Thus taking n such that ⌧  n, it follows that (1.6.1) holds.
Conversely, suppose (1.6.1) is true. To show that M is a martingale, suffices to show
that for n 0, A 2 Fn ,
E[Mn+1 1A ] = E[Mn 1A ].
(1.6.2)
E[Mn+1 1A + Mn 1Ac ] = E[M0 ].
(1.6.3)
F
T
Let ⌧ be defined by ⌧ = (n + 1)1A + n1Ac . Since A 2 Fn , it is easy to check that ⌧ is a
stop time. Using E[M⌧ ] = E[M0 ], it follows that
(1.6.4)
Likewise, using the (1.6.1) for ⌧ = n, we get E[Mn ] = E[M0 ], or equivalently
E[Mn 1A + Mn 1Ac ] = E[M0 ].
A
Now (1.6.2) follows from (1.6.3) and (1.6.4) completing the proof.
Doob’s Maximal Inequality
R
1.7
D
We will now derive an inequality for martingales known as Doob’s maximal inequality. It
plays a major role in stochastic calculus as we will see later.
Theorem 1.25. Let M be a martingale or a positive submartingale. Then, for > 0,
n 1 one has
1
P( max |Mk | > )  E[|Mn |1{max0kn |Mk |> } ].
(1.7.1)
0kn
Further, for 1 < p < 1, there exists a universal constant Cp depending only on p such that
E[( max |Mk |)p ]  Cp E[|Mn |p ].
0kn
(1.7.2)
Proof. Under the assumptions, the process N defined by Nk = |Mk | is a positive submartingale. Let
⌧ = inf{k : Nk > }.
1.7. Doob’s Maximal Inequality
21
Here, and in what follows, we take infimum of the empty set as 1. Then ⌧ is a stop time
and further, {max0kn Nk  } ✓ {⌧ > n}. By Theorem 1.22, the process S defined by
Sk = Nk Nk^⌧ is a submartingale. Clearly S0 = 0 and hence E[Sn ] 0. Note that ⌧ n
implies Sn = 0. Thus, Sn = Sn 1{max0kn Nk > } . Hence
E[Sn 1{max0kn Nk > } ]
0.
This yields
E[N⌧ ^n 1{max0kn Nk > } ]  E[Nn 1{max0kn Nk > } ].
implies ⌧  n and N⌧ > , it follows that
T
Noting that max0kn Nk >
(1.7.3)
N⌧ ^n 1{max0kn Nk >
1{max0kn Nk > } ,
(1.7.4)
F
}
combining (1.7.3) and (1.7.4) we conclude that (1.7.1) is valid.
A
The conclusion (1.7.2) follows from (1.7.1). To see this, fix 0 < ↵ < 1 and let us write
f = (max0kn |Mk |) ^ ↵ and g = |Mn |. Then the inequality (1.7.1) can be rewritten as
R
P(f > ) 
1
E[g 1{f > } ].
(1.7.5)
D
Now consider the product space (⌦, F, P) ⌦ ((0, ↵], B((0, ↵]), µ) where µ is the Lebesgue
measure on (0, ↵]. Consider the function h : ⌦ ⇥ (0, ↵] 7! (0, 1) defined by
h(!, t) = ptp
1
1{t<f (!)} , (!, t) 2 ⌦ ⇥ (0, ↵].
First, note that
Z Z
[
⌦
h(!, t)dt]dP(!) =
(0,↵]
Z
[f (!)]p dP(!)
(1.7.6)
⌦
p
= E[f ].
On the other hand, using Fubini’s theorem in the first and fourth step below and using the
22
Chapter 1. Discrete Parameter Martingales
(1.7.7)
T
estimate (1.7.5) in the second step, we get
Z Z
Z
Z
[
h(!, t)dt]dP(!) =
[ h(!, t)dP(!)]dt
⌦ (0,↵]
(0,↵] ⌦
Z
=
pt(p 1) P(f > t)dt
(0,↵]
Z
1

pt(p 1) E[g 1{f >t} ]dt
t
(0,↵]
Z
Z
(p 2)
=
pt
[ g(!)1{f (!)>t} dP(!)]dt
(0,↵]
⌦
Z Z
= [
pt(p 2) 1{f (!)>t} dt]g(!)dP(!)
⌦ (0,↵]
Z
p
=
g(!)f (p 1) (!)dP(!)
(p 1) ⌦
p
=
E[gf (p 1) ].
(p 1)
p
(p
1)
p
E[gf (p
A
E[f p ] 
F
The first step in (1.7.8) below follows from the relations (1.7.6)-(1.7.7) and the next one
from Holder’s inequality, where q = (p p 1) so that p1 + 1q = 1:
=
1)
p
(p
R
=
(p
1)
1)
1
]
(E[g p ]) p (E[f (p
1
1)q
1
]) q
(1.7.8)
1
(E[g p ]) p (E[f p ]) q .
Since f is bounded, (1.7.8) implies
D
E[f p ]
(1
1
)
q

p
(p
1
1)
(E[g p ]) p
which in turn implies (recalling definitions of f, g), writing Cp = ( (p p 1) )p
E[(( max |Mk |) ^ ↵)p ]  Cp E[|Mn |p ].
0kn
(1.7.9)
Since (1.7.9) holds for all ↵, taking limit as ↵ " 1 (via integers) and using monotone
convergence theorem, we get that (1.7.2) is true.
1.8
Martingale Convergence Theorem
Martingale convergence theorem is one of the key results on martingales. We begin this
section with an upcrossings inequality - a key step in its proof. Let {an : 1  n  m} be
1.8. Martingale Convergence Theorem
23
a sequence of real numbers and ↵ < be real numbers. Let sk , tk be defined (inductively)
as follows: s0 = 0, t0 = 0, and for k = 1, 2, . . . m
sk = inf{n
tk
1
: an  ↵}, tk = inf{n
}.
sk : an
Recall our convention- infimum of an empty set is taken to be 1. It is easy to see that if
tk = j < 1, then
0  s1 < t1 < . . . < sk < tk = j < 1
(1.8.1)
and for i  j
asi  ↵, ati
.
(1.8.2)
We define
Um ({aj : 1  j  m}, ↵, ) = max{k : tk  m}.
m
X
j=1
(ctj ^m
csj ^m )
F
T
Um ({aj : 1  j  m}, ↵, ) is the number of upcrossings of the interval (↵, ) by the
sequence {aj : 1  j  m}. The equations (1.8.1) and (1.8.2) together also imply
tk > (2k 1) and also writing cj = max(↵, aj ) that
↵)Um ({aj }, ↵, ).
(
A
This inequality follows because each completed upcrossings contributes at least
the sum, one term could be non-negative and rest of the terms are zero.
R
Lemma 1.26. For a sequence {an : n
if and only if
lim Um ({aj : 1  j  m}, ↵, ) < 1, 8↵ < , ↵,
D
(1.8.4)
n!1
m!1
Proof. If lim inf n!1 an < ↵ <
↵ to
1} of real numbers,
lim inf an = lim sup an
n!1
(1.8.3)
rationals.
(1.8.5)
< lim supn!1 an then
lim Um ({aj : 1  j  m}, ↵, ) = 1.
m!1
Thus (1.8.5) implies (1.8.4). The other implication follows easily.
It follows that if (1.8.5) holds, then limn!1 an exists in R̄ = R [ { 1, 1}. The next
result gives an estimate on expected value of
Um ({Xj : 1  j  m}, ↵, )
for a submartingale X.
24
Chapter 1. Discrete Parameter Martingales
Theorem 1.27. (Doob’s upcrossings inequality). Let X be a submartingale. Then
for ↵ <
E[|Xm | + |↵| ]
E[Um ({Xj : 1  j  m}, ↵, )] 
.
(1.8.6)
↵
Proof. Fix ↵ <
and define
k
= {n
0
= 0 = ⌧0 and for k
⌧k
1
: Xn  ↵}, ⌧k = {n
1
k
: Xn
Then for each k, k and ⌧k are stop times. Writing Yk = (Xk
sub- martingale and as noted in (1.8.3)
m
X
j=1
(Y⌧j ^m
Y
j ^m
)
}.
↵)+ , we see that Y is also
↵)Um ({Xj : 1  j  m}, ↵, ).
(
(1.8.7)
T
On the other hand, using 0  ( 1 ^ m)  (⌧1 ^ m)  . . .  ( m ^ m)  (⌧m ^ m) = m we
have
m
m
X
X1
Ym Y 1 ^m =
(Y⌧j ^m Y j ^m ) +
(Y j+1 ^m Y⌧j ^m ).
(1.8.8)
Since
have
j+1
^m
j=1
F
j=1
⌧j ^ m are stop times and Y is a submartingale, using Corollary 1.23 we
j+1 ^m
Y⌧j ^m ]
A
E[Y
0.
(1.8.9)
Putting together (1.8.7), (1.8.8) and (1.8.9) we get
Since Yk
0 we get
Y
1 ^m
]
E[(
R
E[Ym
↵)Um ({Xj : 1  j  m}, ↵, )].
D
E[Um ({Xj : 1  j  m}, ↵, )] 
The inequality (1.8.6) now follows using Ym = (Xm
E[Ym ]
.
↵
(1.8.10)
(1.8.11)
↵)+  |Xm | + |↵|.
We recall the notion of uniform integrability of a class of random variables and related
results.
A collection {Z↵ : ↵ 2 } of random variables is said to be uniformly integrable if
lim [sup E[|Z↵ |1{|Z↵ |
K!1 ↵2
K} ]]
= 0.
(1.8.12)
Here are some exercises on Uniform Integrability.
Exercise 1.28. If {Xn : n
1} is uniformly integrable and |Yn |  |Xn | for each n then
show that {Yn : n 1} is also uniformly integrable.
1.8. Martingale Convergence Theorem
Exercise 1.29. Let {Z↵ : ↵ 2
25
} be such that for some p > 1
sup E[|Z↵ |p ] < 1.
↵
show that {Z↵ : ↵ 2
} is Uniformly integrable.
Exercise 1.30. Let {Z↵ : ↵ 2
} be Uniformly integrable. Show that
(i) sup↵ E[|Z↵ | ] < 1.
(ii) 8✏ > 0 9 > 0 such that P(A) <
implies
E[1A |Z↵ | ] < ✏
Hint: For (ii), use E[1A |Z↵ | ]  K + E[|Z↵ |1{|Z↵ |
K} ].
} satisfies (i), (ii) in Exercise 1.30 if and only if
T
Exercise 1.31. Show that {Z↵ : ↵ 2
{Z↵ : ↵ 2 } is uniformly integrable.
(1.8.13)
F
Exercise 1.32. Suppose {X↵ : ↵ 2 } and {Y↵ : ↵ 2 } are uniformly integrable and for
each ↵ 2 , let Z↵ = X↵ + Y↵ . Show that {Z↵ : ↵ 2 } is also uniformly integrable.
The following result on uniform integrability is standard.
Lemma 1.33. Let Z, Zn 2 L1 (P) for n
1.
A
(i) Zn converges in L1 (P) to Z if and only if it converges to Z in probability and {Zn }
is uniformly integrable.
} be a collection of sub -fields of F. Then
R
(ii) Let {G↵ : ↵ 2
{E[Z | G↵ ] : ↵ 2
} is uniformly integrable.
Exercise 1.34. Prove Lemma 1.33. For (i), use Exercise 1.31.
D
We are now in a position to prove the basic martingale convergence theorem.
Theorem 1.35. (Martingale Convergence Theorem) Let {Xn : n
martingale such that
sup E[|Xn | ] = K1 < 1.
n
0} be a sub(1.8.14)
Then the sequence of random variables Xn converges a.e. to a random variable ⇠ with
E[|⇠| ] < 1. Further if {Xn } is uniformly integrable, then Xn converges to ⇠ in L1 (P). If
{Xn } is a martingale or a positive submartingale and if for 1 < p < 1
sup E[|Xn |p ] = Kp < 1,
n
then Xn converges to ⇠ in Lp (P).
(1.8.15)
26
Chapter 1. Discrete Parameter Martingales
Proof. The upcrossings inequality, Theorem 1.27 gives
K1 + |↵|
E[Um ({Xj }, ↵, )] 
↵
for any ↵ < and hence by monotone convergence theorem
E[sup Um ({Xj }, ↵, )] < 1.
m 1
Let N↵ = {supm
1 Um ({Xj }, ↵,
) = 1}. Then P(N↵ ) = 0 and hence if
N ⇤ = [{N↵ : ↵ < , ↵,
rationals}
then P(N ⇤ ) = 0. Clearly, for ! 62 N ⇤ one has
sup Um ({Xj (!)}, ↵, ) < 1 8↵ < , ↵,
rationals.
m 1
T
Hence by Lemma 1.26, for ! 62 N ⇤
⇠ ⇤ (!) = lim inf Xn (!) = lim sup Xn (!).
n!1
= 0 for ! 2
N ⇤,
n!1
by Fatou’s lemma we get
F
Defining
⇠ ⇤ (!)
E[|⇠ ⇤ | ] < 1
so that P(⇠ ⇤ 2 R) = 1. Then defining ⇠(!) = ⇠ ⇤ (!)1{|⇠⇤ (!)|<1} we get
A
Xn ! ⇠ a.e.
R
If {Xn } is uniformly integrable, the a.e. convergence also implies L1 (P) convergence.
If {Xn } is a martingale or a positive submartingale, then by Doob’s maximal inequality,
E[( max |Xk |)p ]  Cp E[|Xn |p ]  Cp Kp < 1
1kn
D
and hence by monotone convergence theorem Z = (supk 1 |Xk |)p is integrable. Now the
convergence in Lp (P) follows from the dominated convergence theorem.
Theorem 1.36. Let {Fm } be an increasing family of sub
1
([1
m=1 Fm ). Let Z 2 L (P) and for 1  n < 1, let
-fields of F and let F1 =
Zn = E[Z | Fn ].
Then Zn ! Z ⇤ = E[Z | F1 ] in L1 (P) and a.s..
Proof. From the definition it is clear that Z is a martingale and it is uniformly integrable
by Lemma 1.33. Thus Zn converges in L1 (P) and a.s. to say Z ⇤ . To complete the proof,
we will show that
Z ⇤ = E[Z | F1 ].
(1.8.16)
1.8. Martingale Convergence Theorem
27
Since each Zn is F1 measurable and Z ⇤ is limit of {Zn } (a.s.), it follows that Z ⇤ is F1
measurable. Fix m 1 and A 2 Fm . Then
E[Zn 1A ] = E[Z 1A ], 8n
m
(1.8.17)
since Fm ✓ Fn for n m. Taking limit in (1.8.17) and using that Zn ! Z ⇤ in L1 (P), we
conclude that 8A 2 [1
n=1 Fn
E[Z ⇤ 1A ] = E[Z 1A ].
(1.8.18)
Since [1
n=1 Fn is a field that generates the -field F1 , the monotone class theorem implies
that (1.8.18) holds for all A 2 F1 and hence (1.8.16) holds.
T
The previous result has an analogue when the -fields are decreasing. Usually one
introduces a reverse martingale (martingale indexed by negative integers) to prove this
result. We avoid it by incorporating the same in the proof.
F
Theorem 1.37. Let {Gm } be a decreasing family of sub -fields of F, i.e. Gm ◆ Gm+1 for
1
all m 1. Let G1 = \1
m=1 Gm . Let Z 2 L (P) and for 1  n  1, let
Zn = E[Z | Gn ].
A
Then Zn ! Z ⇤ in L1 (P) and a.s..
j)
: 1  j  m} is a martingale and the upcrossings
R
Proof. Fix m. Then {(Zm j , Fm
inequality Theorem 1.27 gives
E[Um ({Zm
j
: 1  j  m}, ↵, )] 
E[|Z0 | + |↵| ]
.
↵
D
and hence proceeding as in Theorem 1.35, we can conclude that there exists N ⇤ ✓ ⌦ with
P(N ⇤ ) = 0 and for ! 62 N ⇤ one has
sup Um ({Zm
m 1
j (!)}, ↵,
) < 1 8↵ < , ↵,
rationals.
Now arguments as in Lemma 1.7.1 imply that
lim inf Zn (!) = lim sup Zn (!).
n!1
n!1
By Jensen’s inequality |Zn |  E[|Z0 | | Gn ]. It follows that {Zn } is uniformly integrable (by
Lemma 1.33) and thus Zn converge a.e. and in L1 (P) to a real-valued random variable Z ⇤ .
Since Zn for n m is Gm measurable, it follows that Z ⇤ is Gm -measurable for every m and
hence Z ⇤ is G1 -measurable.
28
Chapter 1. Discrete Parameter Martingales
Also for all A 2 G1 , for all n
Z
1 we have
Zn 1A dP =
Z
Z0 1A dP
since G1 ✓ Gn . Now L1 (P) convergence of Zn to Z ⇤ implies that for all A 2 G1
Z
Z
⇤
Z 1A dP = Z0 1A dP.
Thus E[Z0 | G1 ] = Z ⇤ .
Exercise 1.38. Let ⌦ = [0, 1], F be the Borel -field on [0, 1] and let P denote the Lebesgue
measure on (⌦, F). Let Q be another probability measure on (⌦, F) absolutely continuous
with respect to P, i.e. satisfying : P(A) = 0 implies Q(A) = 0. For n 1 let
n
X (!) =
2
X
2n Q((2
n
(j
1), 2
n
j])1(2
j=1
(!).
T
n
n (j
1),2
n j]
F
Let Fn be the field generated by the intervals {(2 n (j 1), 2 n j] : 1  j  2n }. Show that
(Xn , Fn ) is a uniformly integrable martingale and thus converges to a random variable ⇠ in
L1 (P). Show that ⇠ satisfies
Z
⇠ dP = Q(A) 8A 2 F.
(1.8.19)
A
A
R
Hint: To show uniform integrability of {Xn : n 1} use Exercise 1.31 along with the fact that
absolute continuity of Q w.r.t. P implies that for all ✏ > 0, 9 > 0 such that P(A) < implies
Q(A) < ✏.
D
Exercise 1.39. Let F be a countably generated -field on a set ⌦, i.e. there exists a
sequence of sets {Bn : n
1} such that F = ({Bn : n
1}). Let P, Q be probability
measures on (⌦, F) such that Q absolutely continuous with respect to P. For n
1, let
Fn = ({Bk : k  n}). Then show that
(i) For each n
1, 9 a partition {C1 , C2 . . . , Ckn } of ⌦ such that
Fn = (C1 , C2 . . . , Ckn ).
(ii) For n
1 let
n
X (!) =
kn
X
j=1
1{P(Cj )>0}
Q(Cj )
1C (!).
P(Cj ) j
Show that (X n , Fn ) is a uniformly integrable martingale on (⌦, F, P).
(1.8.20)
1.9. Square Integrable Martingales
29
(iii) X n converges in L1 (P) and also P almost surely to X satisfying
Z
X dP = Q(A) 8A 2 F.
(1.8.21)
A
The random variable X in (1.8.19) and (1.8.20) is called the Radon-Nikodym derivative
of Q w.r.t. P.
A
Square Integrable Martingales
F
1.9
T
Exercise 1.40. Let F be a countably generated -field on a set ⌦. Let be a non-empty
set, A be a -field on . For each 2 let P and Q be probability measures on (⌦, F) such
that Q is absolutely continuous with respect to P . Suppose that for each A 2 F, 7! P (A)
and 7! Q (A) are measurable. Show that there exists ⇠ : ⌦ ⇥ 7! [0, 1) such that ⇠ is
measurable w.r.t F ⌦ A and
Z
⇠(·, )dP = Q (A) 8A 2 F, 8 2
(1.8.22)
Martingales M such that
E[|Mn |2 ] < 1, n
0
(1.9.1)
A
are called square integrable martingales and they play a special role in the theory of stochastic integration as we will see later. Let us note that for p = 2, the constant Cp appearing
in (1.7.2) equals 4. Thus for a square integrable martingale M , we have
E[( max |Mk |)2 ]  4E[|Mn |2 ].
R
(1.9.2)
0kn
D
As seen earlier, Xn = Mn2 is a submartingale and the compensator of X- namely the
predictable increasing process A such that Xn An is a martingale is given by A0 = 0 and
for n 1,
n
X
An =
E[(Xk Xk 1 ) | Fk 1 ].
k=1
The compensator A is denoted as hM, M i. Using
E[(Mk
Mk
1)
2
| Fk
1]
= E[(Mk2
2Mk Mk
= E[(Mk2
Mk2 1 ) | Fk
1
+ Mk2 1 ) | Fk
1]
1]
(1.9.3)
it follows that the compensator can be described as
hM, M in =
n
X
k=1
E[(Mk
Mk
1)
2
| Fk
1]
(1.9.4)
30
Chapter 1. Discrete Parameter Martingales
Thus hM, M i is the unique predictable increasing process with hM, M i0 = 0 such that
Mn2 hM, M in is a martingale. Let us also define another increasing process [M, M ]
associated with a martingale M : [M, M ]0 = 0 and
[M, M ]n =
n
X
(Mk
Mk
1)
2
.
(1.9.5)
k=1
The process [M, M ] is called the quadratic variation of M and the process hM, M i is called
the predictable quadratic variation of M . It can be easily checked that
Mn2
[M, M ]n = 2
n
X
Mk
1 (Mk
Mk
1)
k=1
T
and hence using Theorem 1.14 it follows that Mn2 [M, M ]n is also a martingale. If M0 = 0,
then it follows that
E[Mn2 ] = E[hM, M in ] = E[[M, M ]n ].
(1.9.6)
F
We have already seen that if U is a bounded predictable sequence then the transform
Z defined by (1.5.2)
n
X
Zn =
Uk (Mk Mk 1 )
A
k=1
is itself a martingale. The next result includes an estimate on the L2 (P) norm of Zn .
R
Theorem 1.41. Let M be a square integrable martingale and U be a bounded predictable
process. Let Z0 = 0 and for n 1, let
Zn =
n
X
Uk (Mk
Mk
1 ).
D
k=1
Then Z is itself a square integrable martingale and further
hZ, Zin =
n
X
[Z, Z]n =
n
X
k=1
Uk2 (hM, M ik
hM, M ik
Uk2 ([M, M ]k
[M, M ]k
1)
(1.9.7)
1 ).
(1.9.8)
k=1
As a consequence
E[ max |
1nN
n
X
k=1
Uk (Mk
Mk
1 )|
2
]  4E[
N
X
k=1
Uk2 ([M, M ]k
[M, M ]k
1 )].
(1.9.9)
1.9. Square Integrable Martingales
31
Proof. Since U is bounded and predictable Zn is square integrable for each n. That Z is
square integrable martingale follows from Theorem 1.14. Since
(Zk
Zk
1)
2
= Uk2 (Mk
Mk
1)
2
(1.9.10)
the relation (1.9.8) follows from the definition of the quadratic variation. Further, taking
conditional expectation given Fk 1 in (1.9.10) and using that Uk is Fk 1 measurable, one
concludes
E[(Zk
Zk
1)
2
| Fk
1]
= Uk2 E[(Mk
Mk
1)
2
| Fk
1]
=
Uk2 E[(Mk2
2Mk Mk
=
Uk2 E[(Mk2
Mk2 1 ) | Fk 1 ]
= Uk2 (hM, M ik
1
+ Mk2 1 ) | Fk
hM, M ik
1]
(1.9.11)
1)
T
and this proves (1.9.7). Now (1.9.9) follows from (1.9.8) and the Doob’s maximal inequality,
Theorem 1.25.
E[ max |
Uk (Mk
Mk
1 )|
2
k=1
]  4E[ [M, M ]N ].
A
1nN
n
X
F
Corollary 1.42. For a square integrable martingale M and a predictable process U
P
bounded by 1, defining Zn = nk=1 Uk (Mk Mk 1 ) we have
Further,
[Z, Z]n  [M, M ]n , n
1.
(1.9.12)
(1.9.13)
R
The inequality (1.9.12) plays a very important role in the theory of stochastic integration as later chapters will reveal. We will obtain another estimate due to Burkholder [6]
on martingale transform which is valid even when the martingale is not square integrable.
D
Theorem 1.43. (Burkholder’s inequality) Let M be a martingale and U be a bounded
predictable process, bounded by 1. Then
n
X
9
P( max |
Uk (Mk Mk 1 )|
)  E[|MN | ].
(1.9.14)
1nN
k=1
Proof. Let
⌧ = inf{n : 0  n  N, |Mn |
4
} ^ (N + 1)
so that ⌧ is a stop time. Since {⌧  N } = {max0nN |Mn |
4 }, using Doob’s maximal
inequality (Theorem 1.25) for the positive submartingale {|Mk | : 0  k  N }, we have
4
P(⌧  N )  E(|MN |).
(1.9.15)
32
Chapter 1. Discrete Parameter Martingales
For 0  n  N let ⇠n = Mn 1{n<⌧ } . By definition of ⌧ , |⇠n | 
the set {⌧ = N + 1}, ⇠n = Mn , 8n  N , it follows that
P( max |
1nN
n
X
Uk (Mk
Mk
1 )|
for 0  n  N . Since on
4
)
k=1
 P( max |
1nN
 P( max |
1nN
n
X
Uk (⇠k
⇠k
1 )|
) + P(⌧  N )
Uk (⇠k
⇠k
1 )|
)+
k=1
n
X
4
k=1
(1.9.16)
E(|MN |).
T
We will now prove that for any predictable sequence of random variables {Vn } bounded
by 1, we have
N
X
|E[
Vk (⇠k ⇠k 1 )]|  E[|MN | ].
(1.9.17)
k=1
E[
N
X
Vk (⇠k
⇠k
F
Let us define M̃k = Mk^⌧ for k 0. Since M̃ is a martingale, E[
Writing Yn = ⇠n M̃n , it follows that
1 )] = E[
Vk (Yk
Yk
k=1 Vk (M̃k
1 )].
M̃k
1 )]
(1.9.18)
Since Yn = Mn 1{n<⌧ } Mn^⌧ = Mn^⌧ 1{⌧ n} = M⌧ 1{⌧ n} , it follows that Yk Yk
M⌧ 1{⌧ (k 1)} M⌧ 1{⌧ k} and thus Yk Yk 1 = M⌧ 1{⌧ =k} . Using (1.9.18), we get
Vk (⇠k
⇠k
R
|E[
N
X
= 0.
k=1
A
k=1
N
X
PN
1 )]|
=|E[
k=1
D
E[
E[
N
X
Vk (Yk
k=1
N
X
k=1
N
X
k=1
|Yk
Yk
Yk
1
=
1 )]|
1| ]
(1.9.19)
|M⌧ |1{⌧ =k} ]
E[|M⌧ |1{⌧ N } ]
E[|MN | ]
where for the last step we have used that |Mn | is a submartingale. We will decompose
{⇠n : 0  n  N } into a martingale {Rn : 0  n  N } and a predictable process
{Bn : 0  n  N } as follows: B0 = 0, R0 = ⇠0 and for 0  n  N
Bn = Bn
1
+ (E[⇠n | Fn
1]
⇠n
1)
1.9. Square Integrable Martingales
33
Rn = ⇠ n
Bn .
By construction, {Rn } is a martingale and {Bn } is predictable. Note that
Bn
Bn
= E[⇠n | Fn
1
1]
⇠n
1
⇠n
1 | Fn 1 ].
and hence
Rn
Rn
1
= (⇠n
⇠n
E[⇠n
1)
As a consequence
E[(Rn
Rn
1)
2
]  E[(⇠n
⇠n
1)
2
].
(1.9.20)
E[ max |
Uk (Bk
Bk
1 )| ]
k=1
 E[
A
1nN
n
X
F
T
For x 2 R, let sgn(x) = 1 for x
0 and sgn(x) = 1 for x < 0, so that |x| =
sgn(x)x. Now taking Vk = sgn(Bk Bk 1 ) and noting that Vk is Gk 1 measurable, we have
E[Vk (Bk Bk 1 )] = E[Vk (⇠k ⇠k 1 )] (since ⇠ = R + B and R is a martingale) and hence
= E[
R
= E[
N
X
k=1
N
X
|(Bk
k=1
N
X
Bk
Vk (Bk
Vk (⇠k
1 )| ]
Bk
⇠k
1 )]
(1.9.21)
1 )]
k=1
 E[ |MN | ]
D
where we have used (1.9.19). Thus
P( max |
1nN
n
X
Uk (Bk
Bk
1 )|
2
k=1
Since R is a martingale and U is predictable, Xn =
and hence Xn2 is a submartingale and hence
P( max |
1nN
n
X
k=1
Uk (Rk
Rk
1 )|
2
) =P( max |
1nN

4
2
E[ |MN | ].
Pn
n
X
k=1
2
E[XN
].
2
)
k=1 Uk (Rk
Uk (Rk
Rk
Rk
2
1 )|
(1.9.22)
1)
is a martingale
2
4
)
(1.9.23)
34
Chapter 1. Discrete Parameter Martingales
Since X is a martingale transform of the martingale R, with U bounded by one, we have
2
E[XN
] = E[[X, X]N ]
 E[[R, R]N ]
 E[
 E[
N
X
(Rk
Rk
1)
2
(1.9.24)
]
k=1
N
X
(⇠k
⇠k
1)
2
],
k=1
where we have used (1.9.20). The estimates (1.9.23) and (1.9.24) together yield
n
N
X
4 X
P( max |
Uk (Rk Rk 1 )|
)  2 E[ (⇠k ⇠k 1 )2 ].
1nN
2
k=1
2x(y
⇠02
1.
N
X
⇠k
1 (⇠k
4
|MN | + 2
4
N
X
(⇠k
R
N
X
⇠k
1)
2
k=1
Now (1.9.25) and (1.9.27) together yield
n
X
P( max |
Uk (Rk
D
1nN
Rk
]
1 )|
k=1
Since ⇠k = Rk + Bk , (1.9.22) and (1.9.28) give
n
X
P( max |
Uk (⇠k ⇠k 1 )|
1nN
k=1
1)
(1.9.26)
Wk (⇠k
⇠k
1)
measurable. Using (1.9.19) and
1
3
E[|MN | ].
4
2
)
)
3
5
E[|MN | ].
E[|MN | ].
Finally, (1.9.16) and (1.9.29) together imply the required estimate (1.9.14).
1.10
and summing
k=1
Note that |Wk |  1 and Wk is Gk
E[
⇠k
1
k=1
A

2
F
k=1
x) for y = ⇠k and x = ⇠k
T
Using the identity (y x)2 = y 2 x2
over k, one gets
N
X
2
(⇠k ⇠k 1 )2 =⇠N
where Wk = 4 ⇠k
(1.9.26), we get
(1.9.25)
k=1
Burkholder-Davis-Gundy inequality
If M is a square integrable martingale with M0 = 0, then we have
E[Mn2 ] = E[hM, M in ] = E[[M, M ]n ].
(1.9.27)
(1.9.28)
(1.9.29)
1.10. Burkholder-Davis-Gundy inequality
35
And Doob’s maximal inequality (1.9.2) yields
E[[M, M ]n ] = E[Mn2 ]
 E[( max |Mk |)2 ]
(1.10.1)
1kn
 4E[[M, M ]n ].
Burkholder and Gundy proved that indeed for 1 < p < 1, there exist constants c1p , c2p such
that
p
p
c1p E[([M, M ]n ) 2 ]  E[( max |Mk |)p ]  c2p E[([M, M ]n ) 2 ].
1kn
F
T
Note that fr p = 2, this reduces to (1.10.1). Davis went on to prove the above inequality
for p = 1. This case plays an important role in the result on integral representation of
martingales that we will later consider. Hence we include a proof for the case p = 1
essentially this is the proof given by Davis [14]
Theorem 1.44. Let M be a martingale with M0 = 0. Then there exist universal constants
c1 , c2 such that for all N 1
1
1
A
c1 E[([M, M ]N ) 2 ]  E[ max |Mk | ]  c2 E[([M, M ]N ) 2 ].
1kN
1
R
Proof. Let us define for n
(1.10.2)
Un1 = max |Mk |
1kn
n
X
D
Un2 = (
(Mk
Mk
1)
2
1
)2
k=1
Wn = max |Mk
1kn
Mk
1|
and Vn1 = Un2 , Vn2 = Un1 . The reason for unusual notation is that we will prove
E[Unt ]  130E[Vnt ],
t = 1, 2
(1.10.3)
and this will prove both the inequalities in (1.10.2). Note that by definition, for all n
Wn  2Vnt , Wn  2Unt t = 1, 2.
1,
(1.10.4)
We begin by decomposing M as Mn = Xn + Yn , where X and Y are also martingales
36
Chapter 1. Discrete Parameter Martingales
defined as follows. For n
1 let
Rn = (Mn
Sn = E[Rn | Fn
Xn =
n
X
1 )1{|(Mn Mn
Mn
(Rk
1 )|>2Wn 1 }
1]
Sk )
k=1
Tn = (Mn
Y n = Mn
1 )1{|(Mn Mn
Mn
1 )|2Wn 1 }
Xn
and X0 = Y0 = 0. Let us note that X is a martingale by definition and hence so is Y . Also
that
n
X
Yn =
(Tk + Sk ).
T
k=1
Mn
F
If |Rn | > 0 then |(Mn Mn 1 )| > 2Wn 1 and so Wn = |(Mn
|Rn | = Wn < 2(Wn Wn 1 ). Thus (noting Wn Wn 1 for all n)
n
X
|Rk |  2Wn
k=1
and using that E[ |Sn | ]  E[ |Rn | ], it also follows that
n
X
E[ |Sk | ]  2E(Wn ).
1 )|
and in turn,
(1.10.5)
A
(1.10.6)
k=1
R
Thus (using (1.10.4)) we have
n
n
X
X
E[ |Rk | +
|Sk | ]  4E(Wn )  8E[Vnt ],
k=1
Xk
1
= Rk Sk , (1.10.7) gives us for all n 1,
n
X
E[ |Xk Xk 1 | ]  8E[Vnt ], t = 1, 2.
D
Since Xk
k=1
Let us define for n
t = 1, 2.
(1.10.7)
k=1
1,
A1n = max |Xk |
1kn
n
X
A2n = (
(Xk
Xk
1)
2
1
)2
k=1
Fn1
= max |Yk |
1kn
n
X
Fn2 = (
(Yk
k=1
Yk
1)
2
1
)2
(1.10.8)
1.10. Burkholder-Davis-Gundy inequality
37
and Bn1 = A2n , Bn2 = A1n , G1n = Fn2 , G2n = Fn1 . Since for 1  j  n, |Xj | 
Pn
k=1 |Sk |, the estimates (1.10.7) - (1.10.8) immediately gives us
Pn
E[Atn ]  8E[Vnt ], E[Bnt ]  8E[Vnt ], t = 1, 2.
Also we note here that for all n
k=1 |Rk |
+
(1.10.9)
1, t = 1, 2
Unt  Atn + Fnt
(1.10.10)
Gtn  Bnt + Vnt .
(1.10.11)
and
Thus, for all n
1, t = 1, 2 using (1.10.9) we conclude
E[Unt ]  8E[Vnt ] + E[Fnt ].
T
(1.10.12)
0
(1.10.13)
F
Now using Fubini’s theorem (as used in proof of Theorem 1.25) it follows that
Z 1
t
E[Fn ] =
P(Fnt > x)dx.
x
= inf{n
A
We now will estimate P(FNt > x). For this we fix t 2 {1, 2} and for x 2 (0, 1), define a
stop time
1 : Vnt > x or Gtn > x or |Sn+1 | > x} ^ N.
P(
Z
1
0
is a stop time. If
< N )  P(VNt > x) + P(GtN > x) + P(
D
and hence
x
x
R
Since Sn+1 is Fn measurable, it follows that
P
VNt > x or GtN > x or N
k=1 |Sn | > x. Thus
P(
x
< N )dx 
=
Z
1
0
Z
P(VNt
E[VNt ]
+
PN
k=1 |Sk |
x
< N then either
> x)
1
> x)dx +
P(GtN > x)dx
0
Z 1
P
+
P( N
k=1 |Sn | > x)dx
0
t
E[GN ]
+ E[
PN
k=1 |Sn | ]
t
 E[VNt ] + E[VNt ] + E[BN
] + 2E[WN ]
where we have used (1.10.11) and (1.10.6) in the last step. Now using (1.10.9), (1.10.4)
give us
Z 1
P( x < N )dx  14E[VNt ].
(1.10.14)
0
38
Chapter 1. Discrete Parameter Martingales
Note that S
x
 x and hence
|Gt x
Gt x
1|
 |T
+ S x|
x
 2W
x
 4V tx
1
1
+x
+x
 5x
and as a result, using Gt x
1
 x, we conclude
Gt x  6x.
Hence in view of (1.10.11), we have
T
t
Gt x  min{BN
+ VNt , 6x}.
(1.10.15)
Let Zn = Yn^ x . Then Z is a martingale with Z0 = 0 and
N
X
E[(Zk
Zk
1)
2
] = E[(G1 x )2 ].
F
2
E[ZN
]=
k=1
A
Further, Zn2 is a positive submartingale and hence
P(F 1x > x) = P( max |Zk | > x)
1kN
R
1
2
 2 E[ZN
]
x
1
= 2 E[(G1 x )2 ].
x
D
On the other hand
P(F 2x > x) = P((
PN
k=1 (Zk
Zk
1)
N
1 X
 2
E[(Zk
x
Zk
1)
2 ) 12
2
> x)
]
k=1
1
E[(ZN )2 ]
x2
1
 2 E[(G2 x )2 ].
x
=
Thus, we have for t = 1, 2, for n
1
P(F tx > x) 
1
E[(Gt x )2 ].
x2
(1.10.16)
1.10. Burkholder-Davis-Gundy inequality
39
 14E[VNt ] + E[36Qt ] + E[36Qt ].
Using the estimate (1.10.9), we have
A
F
1
t
E[Qt ] = (E[BN
] + E[VNt ])
6
9
 E[VNt ]
6
and putting together (1.10.17)-(1.10.18) we conclude
E[FNt ]  122E[VNt ]
R
and along with (1.10.12) we finally conclude, for t = 1, 2
t
E[UN
]  130E[VNt ].
D
(1.10.17)
T
t + V t ), we have
Now, writing Qt = 16 (BN
N
Z 1
E[FNt ] =
P(FNt > x)dx
Z0 1
Z 1

P( x < N )dx +
P(F tx > x)dx
0
0
Z 1
1
t
 14E[VN ] +
E[(Gt x )2 ]dx
x2
0
Z 1
1
t
 14E[VNt ] +
E[(min{BN
+ VNt , 6x})2 ]dx
2
x
0
Z 1
36
t
 14E[VN ] + E[
(min{Qt , x})2 dx]
2
x
0
Z Qt
Z 1
36 2
36 t 2
 14E[VNt ] + E[
(x)
dx]
+
E[
(Q ) ]dx
2
2
x
x
t
0
Q
(1.10.18)
Chapter 2
Continuous Time Processes
Notations and Basic Facts
A
2.1
F
T
In this chapter, we will give definitions, set up notations that will be used in the rest of the
book and give some basic results. While some proofs are included, several results are stated
without proof. The proofs of these results can be found in standard books on stochastic
processes.
R
E will denote a complete separable metric space, C(E) will denote the space of real-valued
continuous functions on E and Cb (E) will denote bounded functions in C(E), B(E) will
denote the Borel -field on E. Rd will denote the d-dimensional Euclidean space and
L(m, d) will denote the space of m ⇥ d matrices. For x 2 Rd and A 2 L(m, d), |x| and kAk
will denote the Euclidean norms of x, A respectively.
D
(⌦, F, P) will denote a generic probability space and B(⌦, F) will denote the class of
real-valued bounded F-measurable functions, When ⌦ = E and F = B(E), we will write
B(E) for real-valued bounded Borel- measurable functions.
Recall that for a collection A ✓ F, (A) will denote the smallest -field which contains
A and for a collection G of measurable functions on (⌦, F), (G) will likewise denote the
smallest - field on ⌦ with respect to which each function in G is measurable.
It is well known and easy to prove that for a complete separable metric space E,
(Cb (E)) = B(E)
For an integer d
1, let Cd = C([0, 1), Rd ) with the ucc topology, i.e.
uniform
convergence on compact subsets of [0, 1). With this topology, Cd is itself a complete
40
2.1. Notations and Basic Facts
41
separable metric space. We will denote a generic element in Cd by ⇣ and coordinate
mappings t on Cd by
t (⇣)
= ⇣(t), ⇣ 2 Cd and 0  t < 1.
It can be shown that
B(Cd ) = (
t
: 0  t < 1).
Exercise 2.1. Let ↵ 2 Dd .
F
(i) Show that for any ✏ > 0, and T < 1, the set
T
A function ↵ from [0, 1) to Rd is said to be r.c.l.l. (right continuous with left limits) if
↵ is right continuous everywhere (↵(t) = limu#t ↵(u) for all 0  t < 1) and such that the
left limit ↵(t ) = limu"t ↵(u) exists for all 0 < t < 1. We define ↵(0 ) = 0 and for t 0,
↵(t) = ↵(t) ↵(t ).
For an integer d 1, let Dd = D([0, 1), Rd ) be the space of all r.c.l.l. functions ↵ from
[0, 1) to Rd .
{t 2 [0, T ] : |( ↵)(t)| > ✏}
A
is a finite set.
(ii) Show that the set {t 2 [0, 1) : |( ↵)(t)| > 0} is a countable set.
(iii) Let K = {↵(t) : 0  t  T } [ {↵(t ) : 0  t  T }. Show that K is compact.
R
The coordinate mappings ✓t on Dd are defined by
✓t (↵) = ↵(t), ↵ 2 Dd and 0  t < 1.
D
The space Dd is equipped with the -field (✓t : 0  t < 1). It can be shown that this
-field is same as the Borel -field for the Skorokhod topology (see Ethier and Kurtz[18] ).
An E-valued random variable X defined on a probability space (⌦, F, P) is a measurable
function from (⌦, F) to (E, B(E)). For such an X and a function f 2 Cb (E), E[f (X)] will
denote the integral
Z
E[f (X)] =
f (X(!))dP(!).
⌦
If there are more than one probability measures under consideration, we will denote it as
EP [f (X)].
An E-valued stochastic process X is a collection {Xt : 0  t < 1} of E-valued random
variables. While one can consider families of random variables indexed by sets other than
42
Chapter 2. Continuous Time Processes
[0, 1), say [0, 1] or even [0, 1] ⇥ [0, 1], unless stated otherwise we will take the index set to
be [0, 1). Sometimes for notational clarity we will also use X(t) to denote Xt .
From now on unless otherwise stated, a process will mean a continuous time stochastic
process X = (Xt ) with t 2 [0, 1) or t 2 [0, T ]. For more details and discussions as well
as proofs of statements given without proof in this chapter, see Breiman [5], Ethier and
Kurtz[18], Ikeda and Watanabe[23] Jacod [25], Karatzas and Shreve [42], Metivier [48],
Meyer [49], Protter [50], Revuz-Yor [51], Stroock and Varadhan [57], Williams [56].
Definition 2.2. Two processes X, Y , defined on the same probability space (⌦, F, P) are
said to be equal (written as X = Y ) if
P(Xt = Yt for all t
0) = 1.
8t
T
More precisely, X = Y if 9⌦0 2 F such that P(⌦0 ) = 1 and
0, 8! 2 ⌦0 , Xt (!) = Yt (!).
F
Definition 2.3. A process Y is said to be a version of another process X (written as X
Y ) if both are defined on the same probability space and if
!
0.
A
P(Xt = Yt ) = 1 for all t
v
v
R
It should be noted that in general, X ! Y does not imply X = Y . Take ⌦ = [0, 1]
with F to be the Borel field and P to be the Lebesgue measure. Let Xt (!) = 0 for all
t, ! 2 [0, 1]. For t, ! 2 [0, 1], Yt (!) = 0 for ! 2 [0, 1], ! 6= t and Yt (!) = 1 for ! = t. Easy
v
to see that X ! Y but P(Xt = Yt for all t 0) = 0.
D
Definition 2.4. Two E-valued processes X, Y are said to have same distribution (written
d
as X = Y ), where X is defined on (⌦1 , F1 , P1 ) and Y is defined on (⌦2 , F2 , P2 ), if for all
m 1, 0  t1 < t2 < . . . < tm , A1 , A2 , . . . , Am 2 B(E)
P1 (Xt1 2 A1 , Xt2 2 A2 , . . . , Xtm 2 Am )
= P2 (Yt1 2 A1 , Yt2 2 A2 , . . . , Ytm 2 Am ).
Thus, two processes have same distribution if their finite dimensional distributions are
v
d
the same. It is easy to see that if X ! Y then X = Y .
Definition 2.5. Let X be an E-valued process and D be a Borel subset of E. X is said to
be D-valued if
P(Xt 2 D 8t) = 1.
2.1. Notations and Basic Facts
43
Definition 2.6. An E-valued process X is said to be a continuous process (or a process
with continuous paths) if for all ! 2 ⌦, the path t 7! Xt (!) is continuous.
Definition 2.7. An E-valued process X is said to be an r.c.l.l. process (or a process with
right continuous paths with left limits) if for all ! 2 ⌦, the path t 7! Xt (!) is right continuous
and admits left limits for all t > 0.
Definition 2.8. An E-valued process X is said to be an l.c.r.l. process (or a process with
left continuous paths with right limits) if for all ! 2 ⌦, the path t 7! Xt (!) is left continuous
on (0, 1) and admits right limits for all t 0.
X=X
X
T
For an r.c.l.l. process X, X will denote the r.c.l.l. process defined by Xt = X(t ),
i.e. the left limit at t, with the convention X(0 ) = 0 and let
F
so that ( X)t = 0 at each continuity point and equals the jump otherwise. Note that by
the above convention
( X)0 = X0 .
v
A
Let X, Y be r.c.l.l. processes such that X ! Y . Then it is easy to see that X = Y .
The same is true if both X, Y are l.c.r.l. processes.
R
Exercise 2.9. Prove the statements in the previous paragraph.
It is easy to see that if X is an Rd -valued r.c.l.l. process or an l.c.r.l. process then
sup |Xt (!)| < 1 8T < 1, 8!.
0tT
D
When X is an Rd -valued continuous process, the mapping ! 7! X· (!) from ⌦ into Cd
is measurable and induces a measure P X 1 on (Cd , B(Cd )). This is so because the Borel
-field B(Cd ) is also the smallest -field with respect to which the coordinate process is
measurable. Likewise, when X is an r.c.l.l. process, the mapping ! 7! X· (!) from ⌦ into Dd
is measurable and induces a measure P X 1 on (Dd , B(Dd )). In both cases, the probability
measure P X 1 is called the distribution of the process X
Definition 2.10. A d-dimensional Brownian Motion (also called a Wiener process) is an
Rd -valued continuous process X such that
(i) P(X0 = 0) = 1.
44
Chapter 2. Continuous Time Processes
(ii) For 0  s < t < 1, the distribution of Xt
co-variance matrix (t s)I, i.e. for u 2 Rd
E[exp{iu · (Xt
Xs is Normal (Gaussian) with mean 0 and
Xs )}] = exp{ (t
s)|u|2 }
(iii) For m 1, 0 = t0 < t1 < . . . < tm , the random variables Y1 , . . . , Ym are independent,
where Yj = Xtj Xtj 1 .
Equivalently, it can be seen that a Rd -valued continuous process X is a Brownian motion
if and only if for m 1, 0  t1 < . . . < tm , u1 , u2 , . . . , um 2 Rd ,
E[exp{i
m
X
j=1
uj · Xtj }] = exp{
m
1 X
min(tj , tk )uj · uk }.
2
j,k=1
T
Remark 2.11. The process X in the definition above is sometimes called a standard Brownian
motion and Y given by
Yt = µt + Xt
is a positive constant, is also called a Brownian motion for any µ and .
F
where µ 2 Rd and
D
R
A
When X is a d-dimensional Brownian motion, its distribution, i.e. the induced measure
µ = P X 1 on (Cd , B(Cd )) is known as the Wiener measure. The Wiener measure was
constructed by Wiener before Kolmogorov’s axiomatic formulation of Probability theory.
The existence of Brownian motion or the Wiener measure can be proved in many di↵erent
ways. One method is to use Kolmogorov’s consistency theorem to construct a process
X satisfying (i), (ii) and (iii) in the definition, which determine the finite dimensional
distributions of X and then to invoke the following criterion for existence of a continuous
v
process X̃ such that X ! X̃.
Theorem 2.12. Let X be an Rd -valued process. Suppose that for each T < 1, 9m >
0, K < 1 , > 0 such that
E[|Xt
Xs |m ]  K|t
s|1+ , 0  s  t  T.
Then there exists a continuous process X̃ such that X
v
! X̃.
Exercise 2.13. Let X be a Brownian motion and let Y be defined as follows: Y0 = 0 and
for 0 < t < 1, Yt = tXs where s = 1t . Show that Y is also a Brownian motion.
Definition 2.14. A Poisson Process (with rate parameter ) is an r.c.l.l. non-negative integer
valued process N such that
2.2. Filtration
45
(i) N0 = 0.
(ii) P(Nt
Ns = n) = exp{
n
t} ( n!t) .
(iii) For m 1, 0 = t0 < t1 < . . . < tm , the random variables Y1 , . . . , Ym are independent,
where Yj = Ntj Ntj 1 .
Exercise 2.15. Let N 1 , N 2 be Poisson processes with rate parameters 1 and 2 respectively.
Suppose N 1 and N 2 are independent. Show that N defined by Nt = Nt1 + Nt2 is also a Poisson
process with rate parameter = 1 + 2 .
2.2
T
Brownian motion and Poisson process are the two most important examples of continuous time stochastic processes and arise in modelling of phenomenon occurring in nature.
Filtration
A
F
As in the discrete time case, it is useful to define for t 0, Gt to be the - field generated
by all the random variables observable up to time t and then require any action to be taken
at time t (an estimate of some quantity or investment decision) should be measurable with
respect to Gt . These observations lead to the following definitions.
R
Definition 2.16. A filtration on a probability space (⌦, F, P) is an increasing family of sub
-fields (F⇧ ) = {Ft : t 0} of F indexed by t 2 [0, 1).
Definition 2.17. A process X is said to be adapted to a filtration (F⇧ ) if for all t
is Ft measurable.
0, Xt
D
We will always assume that the underlying probability space is complete i.e. N 2 F,
P(N ) = 0 and N1 ✓ N implies N 2 F and (ii) that F0 contains all sets N 2 F with
P(N ) = 0.
Note that if X is (F⇧ )-adapted and Y = X (see Definition 2.2) then Y is also (F⇧ )adapted in view of the assumption that F0 contains all null sets.
Given a filtration (F⇧ ), we will denote by (F⇧+ ) the filtration {Ft+ : t 0} where
Ft+ = \u>t Fu .
Let N be the class of all null sets (sets with P(N ) = 0), and for a process Z, possibly
vector-valued, let
FtZ = (N [ (Zu : 0  u  t)).
(2.2.1)
46
Chapter 2. Continuous Time Processes
Then (F⇧Z ) is the smallest filtration such that F0 contains all null sets with respect to
which Z is adapted.
While it is not required in the definition, in most situations, the filtration (F⇧ ) under
consideration would be chosen to be (for a suitable process Z)
(F⇧Z ) = {FtZ : t
0}
Sometimes, a filtration is treated as a mere technicality, specially in continuous time
setting as a necessary detail just to define stochastic integral. We would like to stress that
it is not so. See discussion in Section 1.4.
2.3
Martingales and Stop Times
T
Let M be a process defined on a probability space (⌦, F, P) and (F⇧ ) be a filtration.
F
Definition 2.18. M is said to be (F⇧ ) martingale if M is (F⇧ ) adapted, Mt is integrable
for all t and for 0  s < t one has
EP [Mt | Fs ] = Ms .
A
Definition 2.19. M is said to be (F⇧ ) submartingale if M is (F⇧ ) adapted, Mt is integrable
for all t and for 0  s < t one has
EP [Mt | Fs ]
Ms .
D
R
When there is only one filtration under consideration, we will drop reference to it and
call M to be a martingale/ submartingale respectively. If M is a martingale and
is
a convex function on R, then Jensen’s inequality implies that (M ) is a submartingale
provided each (Mt ) is integrable. In particular, if M is a martingale with E[Mt2 ] < 1
for all t then M 2 is a submartingale. We are going to be dealing with martingales that
have r.c.l.l. paths. The next result shows that under minimal conditions on the underlying
filtration one can assume that every martingale has r.c.l.l. paths.
Theorem 2.20. Suppose that the filtration (F⇧ ) satisfies
N0 ✓ N, N 2 F, P(N ) = 0 ) N0 2 F0
\t>s Ft = Fs 8s
0.
(2.3.1)
(2.3.2)
Then every martingale M admits an r.c.l.l. version M̃ i.e. there exists an r.c.l.l. process
M̃ such that
P(Mt = M̃t ) = 1 8t 0.
(2.3.3)
2.3. Martingales and Stop Times
47
Proof. For k, n
1, let snk = k2 n and Xkn = Msnk and Gkn = Fsnk . Then for fixed n,
{Xkn : k
0} is a martingale w.r.t. the filtration {Gkn : k
0}. Thus the Doob’s
upcrossings inequality Theorem 1.27 yields, for rationals ↵ < and an integer T , writing
T n = T 2n ,
E[|MT | ] + |↵|
E[UTn ({Xkn : 0  k  Tn }, ↵, )] 
.
↵
Thus, using the observation that
UTn ({Xkn : 0  k  Tn }, ↵, )  UTn+1 ({Xkn+1 : 0  k  Tn+1 }, ↵, )
we get by the monotone convergence theorem,
E[sup UTn ({Xkn : 0  k  Tn }, ↵, )] 
n 1
E[|MT | ] + |↵|
.
↵
T
Thus we conclude that there exists a set N with P(N ) = 0 such that for ! 62 N ,
sup UTn ({Xkn : 0  k  Tn }, ↵, ) < 1
n 1
(2.3.4)
F
for all rationals ↵ < and integers T . Thus if {tk : k 1} are diadic rationals increasing
or decreasing to t, then for ! 62 N , Mtk (!) converges. Thus defining ✓k (t) = ([t2k ] + 1)2 k ,
where [r] is the largest integer less than or equal to r (the integer part of r), and for ! 62 N ,
M̃t (!) = lim M✓k (t) (!), 0  t < 1
A
k!1
(2.3.5)
R
it follows that t 7! M̃t (!) is a r.c.l.l. function and agrees with Mt (!) for diadic rationals t.
Let us define M̃t (!) = 0 for ! 2 N . Since F0 contains all P null sets, it follows that M̃ is
an r.c.l.l. adapted process and by definition
P(Mt = M̃t ) = 1 for diadic rationals t.
D
Fix t and let tn = ✓n (t). Then \n Ftn = Ft and M̃tn = Mtn converges to M̃t a.s. by
construction. Further,
Mtn = E[Mt+1 | Ftn ] ! E[Mt+1 | Ft ] = Mt a.s.
by Theorem 1.37. Hence (2.3.3) follows.
The conditions (2.3.1) and (2.3.2) together are known as usual Hypothesis in the literature. We will assume (2.3.1) but will not assume (2.3.2). We will mostly consider
martingales with r.c.l.l. paths, and refer to it as an r.c.l.l. martingale.
When we are having only one filtration under consideration, we will drop reference to
it and simply say M is a martingale. We note here an important property of martingales
that would be used later.
48
Chapter 2. Continuous Time Processes
Theorem 2.21. Let M n be a sequence of martingales on some probability space (⌦, F, P)
w.r.t. a fixed filtration (F⇧ ). Suppose that
Mtn ! Mt in L1 (P), 8t
0.
Then M is also a martingale w.r.t. the filtration (F⇧ ).
Proof. Note that for any X n converging to X in L1 (P), for any -field G, using (i), (ii),
(iii) and (iv) in Proposition 1.2, one has
E[X | G]| ] = E[|E[(X n
 E[E[|X n
= E[|X
! 0.
For s < t, applying this to X n = Mtn , one gets
n
X) | G]| ]
X| | G]]
X| ]
T
E[|E[X n | G]
Msn = E[Mtn | Fs ] ! E[Mt | Fs ] in L1 (P).
F
Since Msn ! Ms in L1 (P), it follows that M is a martingale.
A
Remark 2.22. It may be noted that in view of our assumption that F0 contains all null sets,
M as in the statement of the previous theorem is adapted.
Here is a consequence of Theorem 1.35 in continuous time.
R
Theorem 2.23. Let M be a martingale such that {Mt : t
0} is uniformly integrable.
1
Then, there exists a random variable ⇠ such that Mt ! ⇠ in L (P).
D
Proof. Here {Mn : n 1} is a uniformly integrable martingale and Theorem 1.35 yields
that Mn ! ⇠ in L1 (P). Similarly, for any sequence tn " 1, Mtn converges to say ⌘ in
L1 (P). Interlacing argument gives ⌘ = ⇠ and hence Mt ! ⇠ in L1 (P).
The Doob’s maximal inequality for martingales in continuous time follows from the
discrete version easily.
Theorem 2.24. Let M be an r.c.l.l. process. If M is a martingale or a positive submartingale then one has
(i) For
> 0, 0 < t < 1
P[ sup |Ms | > ] 
0st
1
E[|Mt | ].
2.3. Martingales and Stop Times
(ii) For 0 < t < 1
49
E[ sup |Ms |2 ]  4E[|Mt |2 ].
0st
Example 2.25. Let (Wt ) denote a one dimensional Brownian motion on a probability space
(⌦, F, P). Then by the definition of Brownian motion, for 0  s  t, Wt Ws has mean zero
and is independent of {Wu , 0  u  s}. Hence (see (1.3.2)) we have
E[Wt
Ws | FsW ] = 0
and hence W is an (F⇧W )-martingale. Likewise,
E[(Wt
Ws )2 | FsW ] = E[(Wt
Ws ) 2 ] = t
s
and using this and the fact that (Wt ) is a martingale, it is easy to see that
and so defining Mt = Wt2
Ws2 | FsW ] = t
s
T
E[Wt2
t, it follows that M is also an (F⇧W )-martingale.
A
F
Example 2.26. Let (Nt ) denote a Poisson process with rate parameter = 1 on a probability
space (⌦, F, P). Then by the definition of Poisson process, for 0  s  t, Nt Ns has mean
(t s) and is independent of {Nu , 0  u  s}. Writing Mt = Nt t, we can see using (1.3.2)
that
E[Mt Ms | FsN ] = 0
and hence M is an (F⇧N )-martingale. Likewise,
Ms )2 | FsM ] = E[(Mt
R
E[(Mt
Ms ) 2 ] = t
s
D
and using this and the fact that (Mt ) is a martingale, it is easy to see that
and so defining Ut = Mt2
E[Mt2
Ms2 | FsN ] = t
s
t, it follows that U is also an (F⇧N )-martingale.
The notion of stop time, as mentioned in Section 1.6, was first introduced in the context
of Markov chains. Martingales and stop times together are very important tools in the
theory of stochastic process in general and stochastic calculus in particular.
Definition 2.27. A stop time with respect to a filtration (F⇧ ) is a mapping ⌧ from ⌦ into
[0, 1] such that for all t < 1,
{⌧  t} 2 Ft .
50
Chapter 2. Continuous Time Processes
If the filtration under consideration is fixed, we will only refer to it as a stop time. Of
course, for a stop time, {⌧ < t} = [n {⌧  t n1 } 2 Ft . For stop times ⌧ and , it is easy
to see that ⌧ ^ and ⌧ _ are stop times. In particular, ⌧ ^ t and ⌧ _ t are stop times.
Example 2.28. Let X be a Rd -valued process with continuous paths adapted to a filtration
(F⇧ ) and let C be a closed set in R. Then
0 : Xt 2 C}
⌧C = inf{t
(2.3.6)
is a stop time. To see this, define open sets Uk = {x : d(x, C) < k1 }. Writing Ar,k = {Xr 2
Uk } and Qt = {r : r rational, 0  r  t},
{⌧C  t} = \1
k=1 [[r2Qt Ar,k ].
(2.3.7)
T
⌧C is called the hitting time of C.
If X is an r.c.l.l. process, then the hitting time ⌧C may not be a random variable and
hence may not be a stop time. Let us define the contact time C by
= min{inf{t
0 : Xt 2 C}, inf{t > 0 : Xt 2 C}}.
F
C
(2.3.8)
With same notations as above, we now have
and thus
C
C
 t} = [\1
k=1 ([r2Qt Ar,k )] [ {Xt 2 C}
is a stop time. If 0 62 C, then
C
(2.3.9)
A
{
= inf{t
C
can also be described as
0 : Xt 2 C or Xt 2 C}.
R
Exercise 2.29. Construct an example to show that this alternate description may be incorrect
when 0 2 C.
1, ⌧ m defined via
D
If ⌧ is a [0, 1)-valued stop time, then for integers m
⌧m = 2
m
([2m ⌧ ] + 1)
(2.3.10)
is also a stop time since {⌧ m  t} = {⌧ m  2 m ([2m t])} = {⌧ < 2 m ([2m t])} and it follows
that {⌧ m  t} 2 (F⇧ ). Clearly, ⌧ m # ⌧ .
One can see that if k is an increasing sequence of stop times ( k  k+1 for all k 1),
then = limk"1 k is also a stop time. However, if k # , may not be a stop time as
seen in the next exercise.
Exercise 2.30. Let ⌦ = { 1, 1} and P be such that 0 < P(1) < 1. Let Xt (1) = Xt ( 1) = 0
for 0  t  1. For t > 1, let Xt (1) = sin(t 1) and Xt ( 1) = sin(t 1). Let
k
= inf{t
0 : Xt
2
k
},
2.3. Martingales and Stop Times
51
= inf{t
0 : Xt > 0}.
Show that FtX = { , ⌦} for 0  t  1 and FtX = { , {1}, { 1}, ⌦} for t > 1, k are stop
times w.r.t. (F·X ) and k # but is not a stop time w.r.t. (F·X ). Note that { < s} 2 FsX
for all s and yet is not a stop time.
Exercise 2.31. Let ⌧ be a [0, 1)-valued random variable and Ft = (⌧ ^ t). Show that if
⌘ is a stopping time for this filtration then ⌘ = (⌘ _ ⌧ ) ^ r for some r 2 [0, 1].
Definition 2.32. For a stochastic process X and a stop time ⌧ , X⌧ is defined via
X⌧ (!) = X⌧ (!) (!)1{⌧ (!)<1} .
T
Remark 2.33. A random variable ⌧ taking countably many values {sj }, 0  sj < 1 8j is
a stop time if and only if {⌧ = sj } 2 Fsj for all j. For such a stop time ⌧ and a stochastic
process X,
1
X
X⌧ =
1{⌧ =sj } Xsj
F
j=1
and thus X⌧ is a random variable (i.e. a measurable function).
R
A
In general, X⌧ may not be a random variable, i.e. may not be a measurable function.
However, if X has right continuous paths X⌧ is a random variable. To see this, given ⌧
that is [0, 1)-valued, X⌧ m is measurable where ⌧ m is defined via (2.3.10) and X⌧ m ! X⌧
and hence X⌧ is a random variable. Finally, for a general stop time ⌧ , ⇠n = X⌧ ^n is a
random variable and hence
X⌧ = (lim sup ⇠n )1{⌧ <1}
n!1
D
is also a random variable.
Lemma 2.34. Let Z be an r.c.l.l. adapted process and ⌧ be a stop time with ⌧ < 1. Let
Ut = Z⌧ ^t and Vt = Z⌧ 1[⌧,1) (t). Then U, V are adapted processes (with respect to the same
filtration (F⇧ ) that Z is adapted to).
Proof. When ⌧ takes finitely many values it is easy to see that U, V are adapted and for
the general case, the proof is by approximating ⌧ by ⌧ m defined via (2.3.10).
Definition 2.35. For a stop time ⌧ with respect to a filtration (F⇧ ), the stopped -field is
defined by
F⌧ = {A 2 ([t Ft ) : A \ {⌧  t} 2 Ft 8t}.
52
Chapter 2. Continuous Time Processes
We have seen that for a right continuous process Z, Z⌧ is a random variable, i.e.
measurable. The next lemma shows that indeed Z⌧ is F⌧ measurable.
Lemma 2.36. Let X be a right continuous (F⇧ )-adapted process and ⌧ be a stop time. Then
X⌧ is F⌧ measurable.
Proof. For a t < 1, we need to show that {X⌧  a} \ {⌧  t} 2 Ft for all a 2 R. Note
that
{X⌧  a} \ {⌧  t} = {Xt^⌧  a} \ {⌧  t}.
As seen in Lemma 2.34, {Xt^⌧  a} 2 Ft and of course, {⌧  t} 2 Ft .
Applying the previous result to the (deterministic) process Xt = t we get
T
Corollary 2.37. Every stop time ⌧ is F⌧ measurable.
Also, we have
F
Corollary 2.38. Let , ⌧ be two stop times. Then {  ⌧ } 2 F⌧ .
Here is another observation.
A
Proof. Let Xt = 1[ (!),1) (t). We have seen that X is r.c.l.l. adapted and hence X⌧ is F⌧
measurable. Note that X⌧ = 1{ ⌧ } . This completes the proof.
R
Lemma 2.39. Let ⌧ be a stop time and ⇠ be F⌧ measurable random variable. Then Z =
⇠ 1[⌧,1) is an adapted r.c.l.l. process.
D
Proof. Note that for any t, {Zt  a} = {⇠  a} \ {⌧  t} if a < 0 and {Zt  a} = ({⇠ 
a} \ {⌧  t}) [ {⌧ > t} if a
0. Since {⇠  a} 2 F⌧ , we have {⇠  a} \ {⌧  t} 2 Ft
c
and also {⌧ > t} = {⌧  t} 2 Ft . This shows Z is adapted. Of course it is r.c.l.l. by
definition.
The following result will be needed repeatedly in later chapters.
Lemma 2.40. Let X be any real-valued r.c.l.l. process adapted to a filtration (F⇧ ). Let
be any stop time. Then for any a > 0, ⌧ defined as follows is a stop time : if = 1 then
⌧ = 1 and if < 1 then
⌧ = inf{t >
: |Xt
X |
a or |Xt
If ⌧ < 1 then ⌧ > and either |X⌧
X | a or |X⌧
the infimum is finite, it is actually minimum.
X |
X |
a}.
(2.3.11)
a. In other words, when
2.3. Martingales and Stop Times
53
Proof. Let a process Y be defined by Yt = Xt Xt^ . Then as seen in Proposition 2.34,
Y is an r.c.l.l. adapted process. For any t > 0, let Qt = (Q \ [0, t]) [ {t}. Note that since
Y has r.c.l.l. paths one has
{! : ⌧ (!)  t} = \1
n=1 ([r2Qt {|Yr (!)|
a
1
n }).
(2.3.12)
A
F
T
To see this, let ⌦1 denote the left hand side and ⌦2 denote the right hand side.
Let ! 2 ⌦1 . If ⌧ (!) < t, then there is an s 2 [0, t) such that |Ys (!)| a or |Ys (!)| a
(recall the convention that Y0 = 0 and here a > 0). In either case, there exists a sequence
of rationals {rk } in (0, t) converging to s and limk |Yrk (!)| a and thus ! 2 ⌦2 . If ⌧ (!) = t
and if |Yt (!)| a then also ! 2 ⌦2 . To complete the proof of ⌦1 ✓ ⌦2 (recall t 2 Qt ),
we will show that ⌧ (!) = t and |Yt (!)| < a implies |Yt (!)| a. If not, i.e. |Yt (!)| < a,
then ⌧ = t implies that there exist sm > t, sm # t such that for each m, |Ysm (!)| a or
|Ysm (!)| a. This implies |Yt (!)| a and this proves ⌦1 ✓ ⌦2 .
For the other part, if ! 2 ⌦2 , then there exist {rn 2 Qt : n 1} such that |Yrn (!)|
a n1 . Since Qt ✓ [0, t], it follows that we can extract a subsequence rnk that converges
to s 2 [0, t]. By taking a further subsequence if necessary, we can assume that either
rnk s 8k or rnk < s 8k and thus |Yrnk (!)| ! |Ys (!)| a in the first case and |Yrnk (!)| !
|Ys (!)| a in the second case. Also, Y0 (!) = 0 and Y0 (!) = 0 and hence s 2 (0, t]. This
shows ⌧ (!)  s  t and thus ! 2 ⌦1 . This proves (2.3.12).
In each of the cases, we see that either |Y⌧ | a or |Y⌧ | a. Since Yt = 0 for t  , it
follows that ⌧ > . This completes the proof.
R
Remark 2.41. Note that ⌧ is the contact time for the set C = [a, 1) for the process Y .
D
Remark 2.42. If X is also a continuous process, the definition (2.3.11) is same as
⌧ = inf{t >
: |Xt
X |
a}.
(2.3.13)
Exercise 2.43. Construct an example to convince yourself that in the definition (2.3.11) of
⌧ , t > cannot be replaced by t
. However, when X is continuous, in (2.3.13), t > can
be replaced by t
.
Here is another result that will be used in the sequel.
Theorem 2.44. Let X be an r.c.l.l. adapted process with X0 = 0. For a > 0, let { i , : i
0} be defined inductively as follows: 0 = 0 and having defined j : j  i, let i+1 = 1 if
i = 1 and otherwise
i+1
= inf{t >
i
: |Xt
X i|
a or |Xt
X i|
a}.
(2.3.14)
54
Then each
Chapter 2. Continuous Time Processes
i
is a stop time. Further, (i) if
< 1, then
i
i
<
i+1
and (ii) limi"1
i
= 1.
Proof. That each i is a stop time and observation (i) follow from Lemma 2.40. Remains
to prove (ii). If for some ! 2 ⌦,
lim
i"1
i (!)
= t0 < 1
then for such an !, the left limit of the mapping s ! Xs (!) at t0 does not exist, a
contradiction. This proves (ii).
Exercise 2.45. Let X be the Poisson process with rate parameter . Let
i 0, i+1 be defined by (2.3.14) with a = 1. For n 1 let
n 1.
n
Show that
(i) Nt (!) = k if and only if
k (!)
t<
= 0 and for
k+1 (!).
T
⌧n =
0
F
(ii) {⌧n : n 1} are independent random variables with P(⌧n > t) = exp{
and t 0.
t} for all n
1
A
Recall that (F⇧+ ) denotes the fight continuous filtration corresponding to the filtration
(F⇧ ) We now show that the hitting time of the interval ( 1, a) by an r.c.l.l. adapted
process is a stop time for the filtration (F⇧+ ).
R
Lemma 2.46. Let Y be an r.c.l.l. adapted process. Let a 2 R and let
⌧ = inf{t
0 : Yt < a}.
(2.3.15)
D
Then ⌧ is a stop time for the filtration (F⇧+ ).
Proof. Note that for any s > 0, right continuity of t 7! Yt (!) implies that
{! : ⌧ (!) < s} = {! : Yr (!) < a for some r rational, r < s}
and hence {⌧ < s} 2 Fs .
Now for any t, for all k
{⌧  t} = \1
n=k {⌧ < t +
1
} 2 Ft+ 1
k
n
and hence {⌧  t} 2 Ft+ .
Remark 2.47. Similarly, it can be shown that the hitting time of an open set by an r.c.l.l.
adapted process is a stop time for the filtration (F⇧+ ).
2.3. Martingales and Stop Times
55
Here is an observation about (F⇧+ ) stop times.
Lemma 2.48. Let
: ⌦ 7! [0, 1] be such that
{ < t} 2 Ft 8t
Then
0.
is a (F⇧+ )-stop time.
Proof. Let t be fixed and let tm = t + 2
m.
Note that for all m
1,
{  t} = \1
n=m { < tn } 2 Ftm .
+
Thus {  t} 2 \1
m=1 Ftm = Ft .
T
Corollary 2.49. Let {⌧m : m 1} be a sequence of (F⇧ )-stop times. Let = inf{⌧m :
m
1} and ⇣ = sup{⌧m : m
1}. Then ⇣ is a (F⇧ )-stop time whereas is (F⇧+ )-stop
time.
F
Proof. The result follows by observing that
{⇣  t} = \m {⌧m  t}
and
A
{ < t} = [m {⌧m < t}.
R
Exercise 2.50. Show that A 2 F⌧ if and only if the process X defined by
Xt (!) = 1A (!)1[⌧ (!),1) (t)
D
is (F⇧ ) adapted.
Lemma 2.51. Let
 ⌧ be two stop times. Then
F ✓ F⌧ .
Proof. Let A 2 F . Then for t
0, A \ {  t} 2 Ft and {⌧  t} 2 Ft and thus
A \ {  t} \ {⌧  t} = A \ {⌧  t} 2 Ft . Hence A 2 F⌧ .
Here is another result on the family of -fields {F⌧ : ⌧ a stop time}.
Theorem 2.52. Let , ⌧ be stop times. Then
{ = ⌧ } 2 F \ F⌧ .
56
Chapter 2. Continuous Time Processes
Proof. Fix 0  t < 1. For n
1 and 1  k  2n let Ak,n = { k2n1 t <
Bk,n = { k2n1 t < ⌧  2kn t}. Note that

k
2n t}
and
n
2
{ = ⌧ } \ {  t} = \1
n=1 [k=1 [Ak,n \ Bk,n ].
This shows { = ⌧ } \ {  t} 2 Ft for all t and thus { = ⌧ } 2 F . Symmetry now implies
{ = ⌧ } 2 F⌧ as well.
Martingales and stop times are intricately related as the next result shows.
Theorem 2.53. Let M be a r.c.l.l. martingale and ⌧ be a bounded stop time, ⌧  T . Then
E[MT | F⌧ ] = M⌧ .
(2.3.16)
F
T
Proof. We have observed that M⌧ is F⌧ measurable. First let us consider the case when ⌧
is a stop time taking finitely many values, s1 < s2 < . . . < sm  T . Let Aj = {⌧ = sj }.
Since ⌧ is a stop time, Aj 2 Fsj . Clearly, {A1 , . . . , Am } forms a partition of ⌦. Let B 2 F⌧ .
Then, by definition of F⌧ it follows that Cj = B \ Aj 2 Fsj . Since M is a martingale,
E[MT | Fsj ] = Msj and hence
E[MT 1Cj ] = E[Msj 1Cj ] = E[M⌧ 1Cj ].
Summing over j we get
A
E[MT 1B ] = E[M⌧ 1B ].
D
R
This proves (2.3.16) when ⌧ takes finitely many values. For the general case, given ⌧  T ,
let
([2n ⌧ ] + 1)
⌧n =
^ T.
2n
Then for each n, ⌧n is a stop time that takes only finitely many values and further the
sequence {⌧n } decreases to ⌧ . By the part proven above we have
E[MT | F⌧n ] = M⌧n .
(2.3.17)
Now given B 2 F⌧ , using ⌧  ⌧n and Lemma 2.51, we have B 2 F⌧n and hence using
(2.3.17), we have
E[MT 1B ] = E[M⌧n 1B ].
(2.3.18)
Now M⌧n converges to M⌧ (pointwise). Further, in view of Lemma 1.33, (2.3.17) implies
that {M⌧n : n
1} is uniformly integrable and hence M⌧n converges to M⌧ in L1 (P).
Thus taking limit as n ! 1 in (2.3.18) we get
E[MT 1B ] = E[M⌧ 1B ].
This holds for all B 2 F⌧ and hence we conclude that (2.3.16) is true.
2.3. Martingales and Stop Times
57
It should be noted that we have not assumed that the underlying filtration is right
continuous. This result leads to the following
Corollary 2.54. Let M be an r.c.l.l. martingale. Let
times. Then
E[M⌧ | F ] = M .
 ⌧  T be two bounded stop
(2.3.19)
Proof. Taking conditional expectation given F in (2.3.16) and using F ✓ F⌧ we get
(2.3.19).
As in the discrete case, here too we have this characterization of martingales via stop
times.
F
T
Theorem 2.55. Let X be an r.c.l.l. (F⇧ ) adapted process with E[|Xt | ] < 1 for all t < 1.
Then X is an (F⇧ )-martingale if and only if for all (F⇧ )-stop times ⌧ taking finitely many
values in [0, 1), one has
E[X⌧ ] = E[X0 ].
(2.3.20)
Further if X is a martingale then for all bounded stop times
one has
(2.3.21)
A
E[X ] = E[X0 ].
R
Proof. Suppose X is a martingale. Then (2.3.20) and (2.3.21) follow from (2.3.16).
On the other hand suppose (2.3.20) is true for stop times taking finitely many values.
Fix s < t and A 2 Fs . To show E[Xt | Fs ] = Xs , suffices to prove that E[Xt 1A ] = E[Xs 1A ].
Now take
⌧ = s 1 A + t1 A c
D
to get
E[X0 ] = E[X⌧ ] = E[Xs 1A ] + E[Xt 1Ac ]
(2.3.22)
and of course taking the constant stop time t, one has
E[X0 ] = E[Xt ] = E[Xt 1A ] + E[Xt 1Ac ].
(2.3.23)
Now using (2.3.22) and (2.3.23) it follows that E[Xs 1A ] = E[Xt 1A ] and hence X is a
martingale.
Corollary 2.56. For an r.c.l.l. martingale M and a stop time , N defined by
Nt = Mt^
is a martingale.
58
Chapter 2. Continuous Time Processes
We also have the following result about two stop times.
Theorem 2.57. Suppose M is an r.c.l.l. (F⇧ )-martingale and
with  ⌧ . Suppose X is an r.c.l.l. (F⇧ ) adapted process. Let
Nt = X
^t (M⌧ ^t
M
and ⌧ are (F⇧ )-stop times
^t ).
Then N is a (F⇧ )-martingale if either (i) X is bounded or, if (ii) E[X 2 ] < 1 and M is
square integrable.
Proof. Clearly, N is adapted. First consider the case when X is bounded. We will show
that for any bounded stop time ⇣ (bounded by T ),
and then invoke Theorem 2.53. Note that
N⇣ =X
^⇣ (M⌧ ^⇣
=X ˜ (M⌧˜
M˜ )
^⇣ )
^ ⇣  ⌧˜ = ⌧ ^ ⇣ are also bounded stop times. Now
F
where ˜ =
M
T
E[N⇣ ] = 0
E[N⇣ ] = E[E[N⇣ | F ˜ ]]
A
= E[E[X ˜ (M⌧˜
M ˜ ) | F ˜ ]]
= E[X ˜ (E[M⌧˜ | F ˜ ]
M ˜ )]
=0
Since
D
R
as E[M⌧˜ | F ˜ ] = M ˜ by part (ii) of Corollary 2.54. This proves the result when X is
bounded. For (ii), approximating X by X n , where Xtn = max{min{Xt , n}, n} and using
(i) we conclude
E[X n^⇣ (M⌧ ^⇣ M ^⇣ )] = 0.
 ⌧ , we can check that
X n^⇣ (M⌧ ^⇣
M
^⇣ )
= X n (M⌧ ^⇣
M
^⇣ )
and hence that
E[X n (M⌧ ^⇣
M
^⇣ )]
= 0.
By hypothesis, X is square integrable and by Doob’s maximal inequality (2.24) is square
integrable and hence
|X |(sup|Ms |)
sT
is integrable. The required result follows using dominated convergence theorem.
2.4. A Version of Monotone Class Theorem
59
Corollary 2.58. Suppose M is a r.c.l.l. (F⇧ )-martingale and
 ⌧ . Suppose U is a F measurable random variable. Let
Nt = U (M⌧ ^t
M
and ⌧ are stop times with
^t ).
Then N is a (F⇧ )-martingale if either (i) U is bounded or, if (ii) E[U 2 ] < 1 and M is
square integrable.
Proof. Let us define a process X as follows:
Xt (!) = U (!)1[
(!),1) (t).
Then X is adapted by Lemma 2.39. Now the result follows from Theorem 2.57.
Here is another variant that will be useful alter.
^t [(M⌧ ^t
Then Y is a (F⇧ )-martingale.
M
^t )(N⌧ ^t
N
^t )
(B⌧ ^t
B
^t )].
A
Yt = X
F
T
Theorem 2.59. Suppose M, N are r.c.l.l. (F⇧ )-martingales, E(Mt2 ) < 1, E(Nt2 ) < 1 and
and ⌧ are (F⇧ )-stop times with  ⌧ . Suppose B, X are r.c.l.l. (F⇧ ) adapted processes
with E(|Bt |) < 1 for all t and X bounded. Suppose that Z is also a martingale where
Zt = Mt Nt Bt . Let
R
Proof. Since X is assumed to be bounded, it follows that Yt is integrable for all t. Once
again we will show that for all bounded stop times ⇣, E[Y⇣ ] = 0. Let ˜ = ^ ⇣  ⌧˜ = ⌧ ^ ⇣.
Note that
E[Y⇣ ] =E[X ˜ [(M⌧˜
M ˜ )(N⌧˜
=E[X ˜ [(M⌧˜ N⌧˜
Z ˜ )]
D
=E[X ˜ (Z⌧˜
M˜ N˜ )
N˜ )
(B⌧˜
(B⌧˜
E[X ˜ M ˜ (N⌧˜
B ˜ )]]
B˜ )
N˜ )
M ˜ (N⌧˜
N˜ )
E[X ˜ N ˜ (M⌧˜
N ˜ (M⌧˜
M ˜ )]]
M ˜ )]]
=0
by Theorem 2.57 since M, N, Z are martingales. This completes the proof.
2.4
A Version of Monotone Class Theorem
The following functional version of the usual monotone class theorem is very useful in
dealing with integrals and their extension. Its proof is on the lines of the standard version
of monotone class theorem for sets. We will include a proof because of the central role this
result will play in our development of stochastic integrals.
60
Chapter 2. Continuous Time Processes
Definition 2.60. A subset A ✓ B(⌦, F) is said to be closed under uniform and monotone
convergence if f 2 B(⌦, F), f n , g n , hn 2 A for n 1 are such that
(i) f n  f n+1 for all n
1 and f n converges to f pointwise
(ii) g n
1 and g n converges to g pointwise
g n+1 for all n
(iii) hn converges to h uniformly
then f, g, h 2 A.
Here is the functional version of Monotone Class Theorem:
(i)
T
Theorem 2.61. Let A ✓ B(⌦, F) be closed under uniform and monotone convergence.
Suppose G ✓ A is an algebra such that
(G) = F.
F
(ii) 9f n 2 G such that f n  f n+1 and f n converges to 1 pointwise.
Then A = B(⌦, F).
R
A
Proof. Let K ✓ B(⌦, F) be the smallest class that contains G and is closed under uniform
and monotone convergence. Clearly, K contains constants and K ✓ A. Using arguments
similar to the usual (version for sets) monotone class theorem we will first prove that K
itself is an algebra. First we show that K is a vector space. For f 2 B(⌦, F), let
K0 (f ) = {g 2 K : ↵f + g 2 K, 8↵,
2 R}.
D
Note that K0 (f ) is closed under uniform and monotone convergence. First fix f 2 G. Since
G is an algebra, it follows that G ✓ K0 (f ) and hence K = K0 (f ). Now fix f 2 K. The
statement proven above implies that G ✓ K0 (f ) and since K0 (f ) is closed under uniform
and monotone convergence, it follows that K = K0 (f ). Thus K is a vector space.
To show that K is an algebra, for f 2 B(⌦, F) let
K1 (f ) = {g 2 K : f g 2 K}.
Since we have shown that K is a vector space containing constants, it follows that
K1 (f ) = K1 (f + c)
for any c 2 R. Clearly, if f 0, K1 (f ) is closed under monotone and uniform convergence.
Given g 2 B(⌦, F), choosing c 2 R such that f = g+c 0, it follows that K1 (g) = K1 (g+c)
2.4. A Version of Monotone Class Theorem
61
is closed under monotone and uniform convergence. Now proceeding as in the proof of K
being vector space, we can show that K is closed under multiplication and thus K is an
algebra.
Let C = {A 2 F : 1A 2 K}. In view of the assumption (ii), ⌦ 2 C. It is clearly a -field
and also B(⌦, C) ✓ K. Since (G) = F, one has (A) = F where
A = { {f  ↵} : f 2 G, ↵ 2 R}
and hence to complete the proof, suffices to show A ✓ C as that would imply F = C and
in turn
B(⌦, F) ✓ K ✓ A ✓ B(⌦, F)
A
F
T
implying K = A = B(⌦, F). Now to show that A ✓ C, fix f 2 G and ↵ 2 R and let
A = {f  ↵} 2 A. Let |f |  M . Let (x) = 1 for x  ↵ and (x) = 0 for x ↵ + 1 and
(x) = 1 x+↵ for ↵ < x < ↵+1. Using Weierstrass’s approximation theorem, we can get
polynomials pn that converge to uniformly on [ M, M ] and hence boundedly pointwise
on [ M, M ]. Now pn (f ) 2 G as G is an algebra and pn (f ) converges boundedly pointwise
to (f ). Thus (f ) 2 K. Since K is an algebra, it follows that ( (f ))m 2 K for all m 1.
Clearly ( (f ))m converges monotonically to 1A . Thus g m = 1 ( (f ))m 2 K increases
monotonically to 1 1A and hence 1 1A 2 K i.e. A 2 C completing the proof.
Here is a useful variant of the Monotone class Theorem.
R
Definition 2.62. A sequence of real-valued functions fn (on a set S) is said to converge
bp
boundedly pointwise to a function f (written as fn ! f ) if there exists number K such that
|fn (u)|  K for all n, u and fn (u) ! f (u) for all u 2 S.
D
Definition 2.63. A class A ✓ B(⌦, F) is said to be bp-closed if
fn 2 A 8n
bp
1, fn ! f implies f 2 A.
If a set A is bp-closed then it is also closed under monotone and uniform limits and thus
we can deduce the following useful variant of the Monotone class Theorem from Theorem
2.61
Theorem 2.64. Let A ✓ B(⌦, F) be bp-closed. Suppose G ✓ A is an algebra such that
(i)
(G) = F.
(ii) 9f n 2 G such that f n  f n+1 and f n converges to 1 pointwise.
62
Chapter 2. Continuous Time Processes
Then A = B(⌦, F).
Here is an important consequence of Theorem 2.64.
Theorem 2.65. Let F be a -field on ⌦ and let Q be a probability measure on (⌦, F).
Suppose G ✓ B(⌦, F) be an algebra such that (G) = F. Further, 9f n 2 G such that
bp
f n ! 1 (where 1 denotes the constant function 1). Then G is dense in L2 (⌦, F, Q).
2.5
T
Proof. Let K denote the closure of G in L2 (⌦, F, Q) and let A be the set of bounded
functions in K. Then G ✓ A and hence by Theorem 2.64 it follows that A = B(⌦, F).
Hence it follows that K = L2 (⌦, F, Q) as every function in L2 (Q) can be approximated by
bounded functions.
The UCP Metric
ducp (X, Y ) =
1
X
2
E[min(1, sup |Xt
0tm
Yt |)].
(2.5.1)
A
m=1
m
F
Let R0 (⌦, (F⇧ ), P) denote the class of all r.c.l.l. (F⇧ ) adapted processes. For processes
X, Y 2 R0 (⌦, (F⇧ ), P), let
Noting that ducp (X, Y ) = 0 if and only if X = Y (i.e. P(Xt = Yt 8t) = 1), it follows that
ducp is a metric on R0 (⌦, (F⇧ ), P). Now ducp (X n , X) ! 0 is equivalent to
Xt | converging to 0 in probability 8T < 1,
R
sup |Xtn
0tT
ucp
! X.
D
also called uniform convergence in probability, written as X n
Remark 2.66. We have defined ducp (X, Y ) when X, Y are real-valued r.c.l.l. processes. We
can similarly define ducp (X, Y ) when X, Y are Rd -valued r.c.l.l. or l.c.r.l. processes. We will
use the same notation ducp in each of these cases.
In the rest of this section, d is a fixed integer and we will be talking about Rd -valued
processes, and |·| will be the Euclidean norm on Rd .
ucp
When ducp (X n , X) ! 0, sometimes we will write it as X n ! X (and thus the two
mean the same thing). Let X, Y 2 R0 (⌦, (F⇧ ), P). Then for > 0 and integers N
1,
observe that
ducp (X, Y )  2
N
+ + P( sup |Xt
0tN
Yt | > )
(2.5.2)
2.5. The UCP Metric
63
and
P( sup |Xt
Yt | > ) 
0tN
2N
ducp (X, Y ).
(2.5.3)
The following observation, stated here as a remark follows from (2.5.2)-(2.5.3).
Remark 2.67. Z n
n n0
ucp
! Z if and only if for all T < 1, " > 0,
P[ sup|Ztn
tT
Zt | >
> 0, 9n0 such that for
] < ".
(2.5.4)
For an r.c.l.l. process Y , given T < 1, " > 0 one can choose K < 1 such that
P[ sup|Yt | > K ] < "
(2.5.5)
tT
ucp
n 1
T
and hence using (2.5.4) it follows that if Z n ! Z, then given T < 1, " > 0 there exists
K < 1 such that
sup P[ sup|Ztn | > K ] < ".
(2.5.6)
tT
F
The following result uses standard techniques from measure theory and functional analysis but a proof is included as it plays an important part in subsequent chapters.
A
Theorem 2.68. The space R0 (⌦, (F⇧ ), P) is complete under the metric ducp .
Proof. Let {X n : n 1} be a Cauchy sequence in ducp metric. By taking a subsequence if
necessary, we can assume without loss of generality that ducp (X n+1 , X n ) < 2 n and hence
R
1 X
1
X
2
m
n=1 m=1
1
X
D
or equivalently
2
m=1
and thus for all m
implies Z < 1 a.s.)
m
1
X
n=1
E[min(1, sup |Xtn+1
Xtn |)] < 1
E[min(1, sup |Xtn+1
Xtn |)] < 1
0tm
0tm
1 one has (using, for a random variable Z with Z
1
X
n=1
[min(1, sup |Xtn+1
Xtn |)] < 1 a.s.
0tm
Note that for a [0, 1)-valued sequence {an },
1. Hence for all m 1 we have
1
X
sup |Xtn+1
n=1 0tm
0, E[Z] < 1
P
n min(an , 1)
Xtn | < 1 a.s.
< 1 if and only if
P
n an
<
64
Chapter 2. Continuous Time Processes
P
Again, noting that for a real-valued sequence {bn }, n |bn+1 bn | < 1 implies that {bn }
is cauchy and hence converges, we conclude that outside a fixed null set (say N ), Xtn
converges uniformly on [0, m] for every m. So we define the limit to be Xt , which is an
r.c.l.l. process. On the exceptional null set N , Xt is defined to be zero. Note that
ducp (X n+k , X n ) 

n+k
X1
ducp (X j+1 , X j )
j=n
n+k
X1
2
j
j=n
2
n+1
.
As a consequence, (using definition of ducp ) we get that for all integers m,
0tm
n+1
.
T
Xtn+k |)]  2m 2
E[min(1, sup |Xtn
Taking limit as k ! 1 and invoking Dominated convergence theorem we conclude
0tm
It follows that for any T < 1, 0 <
n+1
.
< 1 we have
1
Xt |) > )  2(T +1) 2
n+1
.
(2.5.7)
A
P( sup |Xtn
0tT
Xt |)]  2m 2
F
E[min(1, sup |Xtn
and hence, invoking Remark 2.67, it follows that X n converges in ucp metric to X. Thus
every Cauchy sequence converges and so the space is complete under the metric ducp .
R
Here is a result that will be useful later.
Theorem 2.69. Suppose Z n , Z are r.c.l.l. adapted processes such that
D
Zn
ucp
! Z.
k
Then there exists a subsequence {nk } such that Y k = Z n satisfies
(i) sup0tT |Ytk
Zt | ! 0 a.s. 8T < 1.
(ii) There exists an r.c.l.l. adapted increasing process H such that
|Ytk |  Ht 8t < 1, 8n
1.
(2.5.8)
Proof. Since ducp (Z n , Z) ! 0, for k
1, we can choose nk with nk+1 > nk such that
k
ducp (Z n , Z)  2 k . Then as seen in the proof of Theorem 2.68, this implies
1
X
[sup|Ytk Zt |] < 1, 8T < 1 a.s.
k=1 tT
2.5. The UCP Metric
65
Hence (i) above holds for this choice of {nk }. Further, defining
Ht = sup [
0st
1
X
k=1
|Ysk
Zs | + |Zs |]
we have that H is an r.c.l.l. adapted increasing process and |Y k |  H for all k
1.
Remark 2.70. If we have two, or finitely many sequences {Z i,n } converging to Z i in ducp ,
i = 1, 2, . . . , p then we can get one common subsequence {nk } and a process H such that (i),
(ii) above hold for i = 1, 2, . . . p. All we need to do is to choose {nk } such that
k
ducp (Z i,n , Z i )  2
k
,
i = 1, 2 . . . p.
T
Exercise 2.71. An alternative way of obtaining the conclusion in Remark 2.70 is to apply
Theorem 2.69 to an appropriately defined Rdp -valued processes.
The following Lemma will play an important role in the theory of stochastic integration.
ucp
! Zt^⌧ m as n " 1.
(2.5.9)
A
n
Zt^⌧
m
F
Lemma 2.72. Let Z n , Z be adapted processes and let ⌧ m be a sequence of stop times such
that ⌧ m " 1. Suppose that for each m
Then
Zn
ucp
! Z as n " 1.
R
Proof. Fix T < 1, " > 0 and ⌘ > 0. We need to show that 9n0 such that for n
P[ sup|Ztn
tT
Zt | > ⌘ ] < ".
n0
(2.5.10)
D
First, using ⌧ m " 1, fix m such that
n
Using Zt^⌧
m
P[ ⌧ m < T ] < "/2.
ucp
! Zt^⌧ m , get n1 such that for n
n
P[ sup|Zt^⌧
m
tT
n1
Zt^⌧ m | > ⌘ ] < "/2.
Now,
{ sup|Ztn
tT
and hence for n
n
Zt | > ⌘} ✓ {{sup|Zt^⌧
m
tT
(2.5.11)
Zt^⌧ m | > ⌘} [ {⌧ m < T }}
n1 , (2.5.10) follows from (2.5.11) and (2.5.12).
The same argument as above also yields the following.
(2.5.12)
66
Chapter 2. Continuous Time Processes
Corollary 2.73. Let Z n be r.c.l.l. adapted processes and let ⌧ m be a sequence of stop
times such that ⌧ m " 1. For n 1, m 1 let
n
Ytn,m = Zt^⌧
m.
Suppose that for each m, {Y n,m : n
in ducp metric.
2.6
1} is Cauchy in ducp metric then Z n is Cauchy in
The Lebesgue-Stieltjes Integral
Let G : [0, 1) 7! R be an r.c.l.l. function. For 0  a < b < 1 the total variation Var[a,b]
of G(s) over [a, b] and Var(a,b] over (a, b] are defined as follows:
sup
[
m
X
|G(tj )
G(tj
T
Var(a,b] (G) =
{ti }:a=t0 <t1 <...<tm =b, m 1 j=1
1 )|].
(2.6.1)
F
Var[a,b] (G) = |G(a)| + Var(a,b] (G).
A
If Var[0,t] (G) < 1 for all t, G will be called a function with finite variation. It is
well known that a function has finite variation paths if and only if it can be expressed as
di↵erence of two increasing functions.
If Var[0,t] (G) < 1 for all t, the function |G|t = Var[0,t] (G) is then an increasing [0, 1)valued function. Let us fix such a function G.
Here,
D
R
For any T fixed, there exists a unique countably additive measure
additive signed measure µ on the Borel -field of [0, T ] such that
and a countably
([0, t]) = |G|(t) 8t  T
(2.6.2)
µ([0, t]) = G(t) 8t  T.
(2.6.3)
is the total variation measure of the signed measure .
R
For measurable function h on [0, T ], if hd < 1, then we define
Z t
Z
|h|s d|G|s = |h|1[0,t] d
(2.6.4)
0
and
Z
Note that we have
|
Z
hs dGs =
Z
hs dGs | 
Z
t
0
t
0
h1[0,t] dµ.
(2.6.5)
t
0
|h|s d|G|s .
(2.6.6)
2.6. The Lebesgue-Stieltjes Integral
67
It follows that if h is a bounded measurable function on [0, 1), then
Rt
hdG is defined
Rt
and further if
h, then the dominated convergence theorem yields that Htn = 0 hn dG
Rt
converges to H(t) = 0 hdG uniformly on compact subsets.
bp
hn !
0
Exercise 2.74. Let G be an r.c.l.l. function on [0, 1) such that |G|t < 1 for all t < 1.
Show that for all T < 1
X
|( G)t | < 1.
(2.6.7)
tT
Note that as seen in Exercise 2.1, {t : |( G)t | > 0} is a countable set and thus the sum
appearing above is a sum of countably many terms.
Hint: Observe that the left hand side in (2.6.7) is less than or equal to |G|T .
F
T
Let us denote by V+ = V+ (⌦, (F⇧ ), P) the class of (F⇧ ) adapted r.c.l.l. increasing
processes A with A0 0 and by V = V(⌦, (F⇧ ), P) the class of r.c.l.l. adapted processes B
such that A defined by
At (!) = Var[0,t] (B· (!))
(2.6.8)
R
A
belongs to V+ . A process A 2 V will be called process with finite variation paths. It is
easy to see that B 2 V if and only if B can be written as di↵erence of two processes in
V+ : indeed, if A is defined by (2.6.8), we have B = D C where D = 12 (A + B) and
C = 12 (A B) and C, D 2 V+ . Let V0 and V+
0 denote the class of processes A in V and
+
V respectively such that A0 = 0. For B 2 V, we will denote the process A defined by
(2.6.8) as A = Var(B).
A process A 2 V will be said to be purely discontinuous if
X
At = A0 +
( A)s .
0<st
D
Exercise 2.75. Show that every A 2 V can be written uniquely as A = B +C with B, C 2 V,
B being a continuous process with B0 = 0 and C being a purely discontinuous process.
Lemma 2.76. Let B 2 V and let X be a bounded l.c.r.l. adapted process. Then
Z t
Ct (!) =
Xs (!)dBs (!)
(2.6.9)
0
is well defined and is an r.c.l.l. adapted process. Further, C 2 V.
Proof. For every ! 2 ⌦, t 7! Xt (!) is a bounded measurable function and hence C is well
defined. For n 1 and i 0 let tni = ni . Let
Xtn = Xtni
for tni  t < tni+1 .
68
Chapter 2. Continuous Time Processes
bp
Then X n ! X. Clearly
Ctn (!)
=
Z
t
0
Xsn (!)dBs (!) =
X
i : tn
i t
Xtni (Btni+1 ^t
Btni ^t )
bp
is r.c.l.l. adapted and further, C n ! C. Thus C is also an r.c.l.l. adapted process. Let
A = Var(B). For any s < t,
Z t
|Ct (!) Cs (!)| 
|Xu (!)|dAs (!)
s
 K(At (!)
As (!)).
T
where K is a bound for the process X. Since A is an increasing process, it follows that
Var[0,T ] (C)(!)  KAT (!) < 1 for all T < 1 and for all !.
F
Exercise 2.77. Show that the conclusion in Lemma 2.76 is true even when X is a bounded
r.c.l.l. adapted process. In this case, Dtn defined by
Z t
X
n
Dt (!) =
Xsn (!)dBs (!) =
Xtni+1 ^t (Btni+1 ^t Btni ^t )
0
0
X dB.
A
Rt
D
R
converges to
i : tn
i t
Chapter 3
The Ito Integral
Quadratic Variation of Brownian Motion
T
3.1
A
F
Let (Wt ) denote a one dimensional Brownian motion on a probability space (⌦, F, P). We
have seen that W is a martingale w.r.t. its natural filtration (F⇧W ) and with Mt = Wt2 t,
M is also (F⇧W ) martingale. These properties are easy consequence of the independent
increment property of Brownian motion.
Wiener and Ito realized the need to give a meaning to limit of what appeared to be
Riemann-Stieltjes sums for the integral
Z t
fs dWs
(3.1.1)
0
D
R
in di↵erent contexts- while in case of Wiener, the integrand was a deterministic function,
Ito needed to consider a random process (fs ) that was a non-anticipating function of W i.e. f is adapted to (F⇧W ).
It is well known that paths s 7! Ws (!) are nowhere di↵erentiable for almost all !
and hence we cannot interpret the integral in (3.1.1) as a path-by-path Riemann-Stieltjes
integral. We will deduce the later result from a weaker result that is relevant for stochastic
integration.
Theorem 3.1. Let (Wt ) be a Brownian motion. Let tni = i2 n , i
0, n
1. Let
P1
P1
n
n
2
n
n
Vt =
Wtni ^t |, Qt =
Wtni ^t ) . Then for all t > 0, (a)
i=0 (Wti+1 ^t
i=0 |Wti+1 ^t
n
n
Vt ! 1 a.s. and (b) Qt ! t a.s.
Proof. We will first prove (b). Let us fix t < 1 and let
Xin = Wtni+1 ^t
Wtni ^t ,
Zin = (Wtni+1 ^t
69
Wtni ^t )2
(tni+1 ^ t
tni ^ t).
70
Chapter 3. The Ito Integral
Then from properties of Brownian motion it follows that {Xin , i
0} are independent
n
n
2
random variables with normal distribution and E(Xi ) = 0, E(Xi ) = (tni+1 ^ t tni ^ t).
So, {Zin , i 0} are independent random variables with E(Zin ) = 0 and E(Zin )2 = 2((tni+1 ^
t tni ^ t))2 . Now
1
X
n
2
E(Qt t) =E(
Zin )2
i=0
E(Zin )2
i=0
1
X
=2
i=0
(tni+1 ^ t
2
n+1
=2
n+1
1
X
i=0
(3.1.2)
tni ^ t)2
(tni+1 ^ t
tni ^ t)
T
=
1
X
t.
P1
F
Note that each of the sum appearing above is actually a finite sum. Thus
1
X
E
(Qnt t)2  t < 1
n=1
t)2
so that
< 1 a.s. and hence Qnt ! t a.s.
For (a), let ↵( , !, t) = sup{|Wu (!) Wv (!)| : |u v|  , u, v 2 [0, t]}. Then uniform
continuity of u 7! Wu (!) implies that for each !, t < 1,
A
n
n=1 (Qt
R
lim ↵( , !, t) = 0.
Now note that for any !,
D
Qnt (!) ( max |Wtni+1 ^t (!)
0i<1
=↵(2
n
(3.1.3)
#0
Wtni ^t (!)|)(
, !, t)Vtn (!).
1
X
i=0
|Wtni+1 ^t
Wtni ^t |)
(3.1.4)
So if lim inf n Vtn (!) < 1 for some !, then lim inf n Qnt (!) = 0 in view of (3.1.3) and (3.1.4).
For t > 0, since Qnt ! t a.s., we must have Vtn ! 1 a.s.
Exercise 3.2. For any sequence of partitions
m
m
0 = tm
0 < t1 < . . . < tn < . . . ;
of [0, 1) such that for all T < 1,
m (T )
=(
sup
{n : tm
n T }
(tm
n+1
tm
n " 1 as n " 1
tm
n )) ! 0 as m " 1
(3.1.5)
(3.1.6)
3.2. Levy’s Characterization of the Brownian motion
let
Qm
t
=
1
X
n=0
(Wtm
n+1 ^t
71
Wtm
)2 .
n ^t
(3.1.7)
Show that for each t, Qm
t converges in probability to t.
Remark 3.3. It is well known that the paths of Brownian motion are nowhere di↵erentiable.
For this and other path properties of Brownian motion, see Breiman [5], McKean [45], Karatzas
and Shreve[42].
i=0
Wsni ^t (Wtni+1 ^t
Wtni ^t )
(3.1.8)
F
1
X
T
Remark 3.4. Since the paths of Brownian motion do not have finite variation on any interval,
R
we cannot invoke Riemann-Stieltjes integration theory for interpreting X dW , where W is
Brownian motion. The following calculation shows that the Riemann-Stieltjes sums do not
R
converge in any weaker sense (say in probability) either. Let us consider W dW . Let tni =
i2 n , i 0, n 1. The question is whether the sums
converge to some limit for all choices of sni such that tni  sni  tni+1 . Let us consider two
cases sni = tni+1 and sni = tni :
1
X
A
Ant =
i=0
=
1
X
R
Btn
Wtni+1 ^t (Wtni+1 ^t
i=0
Wtni ^t (Wtni+1 ^t
Wtni ^t )
(3.1.9)
Wtni ^t ).
(3.1.10)
D
Now Ant and Btn cannot converge to the same limit as their di↵erence satisfies
(Ant
Btn ) = Qnt .
Thus even in this simple case, the Riemann-Stieltjes sums do not converge to a unique limit.
In this case, it is possible to show that Ant and Btn actually do converge but to two di↵erent
limits. It is possible to choose {sni } so that the Riemann sums in (3.1.8) do not converge.
3.2
Levy’s Characterization of the Brownian motion
Let (Wt ) be a Wiener process and let (Ft ) be a filtration such that W is a martingale w.r.t.
(Ft ). Further suppose that for every s,
{Wt
Ws : t
s} is independent of Fs .
(3.2.1)
72
Chapter 3. The Ito Integral
R
A
F
T
It follows that Mt = Wt2 t is also a martingale w.r.t (Ft ). Levy had proved that if W is
any continuous process such that both W, M are (F⇧ )-martingales then W is a Brownian
motion and (3.2.1) holds. Most proofs available in texts today deduce this as an application
of Ito’s formula. We will give an elementary proof of this result which uses interplay of
martingales and stop times. The proof is motivated by the proof given in Ito’s lecture notes
[24], but the same has been simplified using partition via stop times instead of deterministic
partitions.
We will use the following inequalities on the exponential function which can be easily
proven using Taylor’s theorem with remainder. For a, b 2 R
1
|eb (1 + b)|  e|b| |b|2
2
ia
|e
1|  |a|
1 2
1
|eia (1 + ia
a )|  |a|3 .
2
6
Using these inequalities, we conclude that for a, b such that |a|  , |b|  , < loge (2),
we have
1 2
|eia+b (1 + ia
a + b)|
2
1 2
 |eia [eb (1 + b)] + b(eia 1) + (eia (1 + ia
a ))|
2
(3.2.2)
1 |b| 2
1 3
 ( e |b| + |b||a| + |a| )
2
6
2
 (2|b| + |a| ).
D
Theorem 3.5. Let X be a continuous process adapted to a filtration (F⇧ ) and let Mt =
Xt2 t for t 0. Suppose that (i) X0 = 0, (ii) X is a (F⇧ )-martingale and (iii) M is a
(F⇧ ) martingale. Then X is a Brownian motion and further, for all s
{(Xt
Xs ) : t
s} is independent of Fs .
Proof. We will prove this in a series of steps.
step 1: For bounded stop times  ⌧ , say ⌧  T ,
E[(X⌧
X )2 ] = E[(⌧
)].
To see this, Corollary 2.58 and the hypothesis that M is a martingale imply that
Yt = X⌧2^t
X 2^t
(⌧ ^ t
^ t)
is a martingale and hence E[YT ] = E[Y0 ] = 0. This proves step 1.
(3.2.3)
3.2. Levy’s Characterization of the Brownian motion
Let us fix
2 R and let
73
1 2
t}.
2
step 2: For each bounded stop time , E[Z ] = 1. This would show that Z is a (F⇧ )martingale. To prove this claim, fix and let  T . Let be sufficiently small such that
(| | + 2 ) < loge (2). Let us define a sequence of stop times {⌧i : i
1} inductively as
follows: ⌧0 = 0 and for i 0,
Zt = exp{i Xt +
⌧i+1 = inf{t
⌧i : |Xt
X⌧i |
or |t
⌧i |
}
or t
(3.2.4)
Inductively, using Theorem 2.44 one can prove that each ⌧i is a stop time and that for each
!, ⌧j (!) = (!) for sufficiently large j. Further, continuity of the process implies
X⌧i |  , |⌧i+1
Let us write
Z ⌧m
1=
m
X1
⌧i |  .
(3.2.5)
T
|X⌧i+1
(Z⌧k+1
Z⌧k ).
(3.2.6)
k=0
E[(Z⌧k+1
Z⌧k )] = E[Z⌧k (ei
F
Note that
(X⌧k+1 X⌧k )+ 12
Since E[X⌧k+1
X⌧k )+ 12
(X⌧k+1
A
= E[Z⌧k E(ei
2 (⌧
k+1
2 (⌧
⌧k )
k+1
1)]
⌧k )
1 | F⌧k )]
X⌧k | F⌧k ] = 0 as X is a martingale and
X⌧k ) 2
R
E[(X⌧k+1
⌧k ) | F⌧k ] = 0
(⌧k+1
(3.2.7)
as seen in step 1 above, we have
E[ei
(X⌧k+1 X⌧k )+ 12
(X⌧k+1
D
= E[ei
2 (⌧
k+1
⌧k )
X⌧k )+ 12
2 (⌧
1 2
(X⌧k+1
2
1 | F⌧k ]
k+1
⌧k )
X⌧k ) 2 +
{1 + i (X⌧k+1
1 2
(⌧k+1
2
X⌧k )
⌧k )} | F⌧k ]
Using (3.2.2), (3.2.5) and the choice of , we can conclude that the expression on the right
hand side inside the conditional expectation is bounded by
2
((X⌧k+1
X⌧k )2 + (⌧k+1
⌧k ))
Putting together these observations and (3.2.7), we conclude that
|E[(Z⌧k+1
Z⌧k )]|  2
2
E[(⌧k+1
As a consequence
|E[Z⌧m
1]|  2
2
E[⌧m ].
⌧k )].
74
Chapter 3. The Ito Integral
1
Now Z⌧m is bounded by e 2
2T
and converges to Z , we conclude
|E[Z
Since this holds for all small
2
1]|  2
T.
> 0 it follows that
E[Z ] = 1
and this completes the proof of step 2.
step 3: For s < t, 2 R,
E[e(i
(Xt Xs ))
1
2
| Fs ] = e
2 (t
s)
.
(3.2.8)
T
We have seen that Zt is a martingale and (3.2.8) follows from it since Xs is Fs measurable.
As a consequence
1 2
E[e(i (Xt Xs )) ] = e 2 (t s) .
(3.2.9)
E[e(i
(Xt Xs )+i✓Y )
F
and so the distribution of Xt Xs is Gaussian with mean 0 and variance (t
step 4:
For s < t, , ✓ 2 R, a Fs measurable random variable Y we have
] = E[e(i
(Xt Xs ))
]E[e(i✓Y ) ].
s).
(3.2.10)
R
A
The relation (3.2.10) is an immediate consequence of (3.2.8). We have already seen that
Xt Xs has Gaussian distribution with mean 0 and variance t s and (3.2.10) implies
that Xt Xs is independent of Fs , in particular, Xt Xs is independent of {Xu : u  s}.
This completes the proof.
D
Let W = (W 1 , W 2 , . . . , W d ) be d-dimensional Brownian motion, where W j is the j th
component. In other words, each W j is a real-valued Brownian motion and W 1 , W 2 , . . . , W d
are independent.
P
For any ✓ = (✓1 , ✓2 , . . . , ✓d ) 2 Rd with |✓| = dj=1 (✓j )2 = 1,
Xt✓ =
d
X
✓j Wtj
(3.2.11)
j=1
is itself a Brownian motion and hence with
Mt✓ = (Xt✓ )2
t,
8✓ 2 Rd with |✓| = 1; X ✓ , M ✓ are (F⇧W )-martingales.
(3.2.12)
(3.2.13)
Indeed, using Theorem 3.5, we will show that (3.2.13) characterizes multidimensional
Brownian motion.
3.3. The Ito Integral
75
Theorem 3.6. Let W be an Rd -valued continuous process such that W0 = 0. Suppose (F⇧ )
is a filtration such that W is (F⇧ ) adapted. Suppose W satisfies
8✓ 2 Rd with |✓| = 1; X ✓ , M ✓ are (F⇧ )-martingales,
(3.2.14)
where X ✓ and M ✓ are defined via (3.2.11) and (3.2.12). Then W is a d-dimensional
Brownian motion and further, for 0  s  t
Ws ) is independent of Fs .
(Wt
Proof. Theorem 3.5 implies that for ✓ 2 Rd with |✓| = 1 and
E[exp{ (✓ · Wt
✓ · Ws )} | Fs ] = exp{
2R
1 2
(t
2
Ws ) and Fs also follows
T
This implies that W is a Brownian motion. Independence of (Wt
as in Theorem 3.5.
s)}.
3.3
The Ito Integral
A
F
Theorems 3.5 and 3.6 are called Levy’s characterization of Brownian motion (one dimensional and multidimensional cases respectively).
R
Let S be the class of stochastic processes f of the form
fs (!) = a0 (!)1{0} (s) +
m
X
aj+1 (!)1(sj ,sj+1 ] (s)
(3.3.1)
j=0
D
where 0 = s0 < s1 < s2 < . . . < sm+1 < 1, aj is bounded Fsj 1 measurable random
variable for 1  j  (m + 1), and a0 is bounded F0 measurable. Elements of S will be
R
called simple processes. For an f given by (3.3.1), we define X = f dW by
Xt (!) =
m
X
j=0
aj+1 (!)(Wsj+1 ^t (!)
Wsj ^t (!)).
(3.3.2)
R
a0 does not appear on the right side because W0 = 0. It can be easily seen that f dW
defined via (3.3.1) and (3.3.2) for f 2 S does not depend upon the representation (3.3.1).
In other words, if g is given by
gt (!) = b0 (!)1{0} (s) +
n
X
j=0
bj+1 (!)1(rj ,rj+1 ] (t)
(3.3.3)
76
Chapter 3. The Ito Integral
where 0 = r0 < r1 < . . . < rn+1 and bj is Frj 1 measurable bounded random variable,
R
R
1  j  (n + 1), and b0 is bounded F0 measurable and f = g, then f dW = gdW i.e.
m
X
j=0
aj+1 (!)(Wsj+1 ^t (!)
=
n
X
j=0
Wsj ^t (!))
(3.3.4)
bj+1 (!)(Wrj+1 ^t (!)
Wrj ^t (!)).
T
Rt
By definition, X is a continuous adapted process. We will denote Xt as 0 f dW . We
will obtain an estimate on the growth of the integral defined above for simple f 2 S and
then extend the integral to an appropriate class of integrands - those that can be obtained
as limits of simple processes. This approach is di↵erent from the one adopted by Ito and
we have adopted this approach with an aim to generalize the same to martingales.
R
We first note some properties of f dW for f 2 S and obtain an estimate.
F
Lemma 3.7. Let f, g 2 S and let a, b 2 R. Then
Z t
Z t
Z t
(af + bg)dW = a
f dW + b
gdW.
0
0
(3.3.5)
0
A
Proof. Let f, g have representations (3.3.1) and (3.3.3) respectively. Easy to see that we
can get 0 = t0 < t1 < . . . < tk such that
R
{tj : 0  j  k} = {sj : 0  j  m} [ {rj : 0  j  n}
and then represent both f, g over common time partition. Then the result (3.3.5) follows
easily.
D
Rt
Rt
Rt
Lemma 3.8. Let f, g 2 S, and let Yt = 0 f dW , Zt = 0 gdW and At = 0 fs gs ds,
Mt = Yt Zt At . Then Y, Z, M are (F⇧ )-martingales.
Proof. By linearity property (3.3.5) and the fact that sum of martingales is a martingale,
suffices to prove the lemma in the following two cases:
Case 1: 0  s < r and
ft = a1(s,r] (t), gt = b1(s,r] (t), a, b are Fs measurable.
Case 2: 0  s < r  u < v and
ft = a1(s,r] (t), gt = b1(u,v] (t), a is Fs measurable and b is Fu measurable.
3.3. The Ito Integral
77
Here in both cases, a, b are assumed to be bounded. In both the cases, Yt = a(Wt^r Wt^s ).
That Y is a martingale follows from Theorem 2.57. Thus in both cases, Y is an (F⇧ )martingale and similarly, so is Z. Remains to show that M is a martingale. In Case 1,
writing Nt = Wt2 t
Mt =ab((Wt^r
2
=ab((Wt^r
=ab(Nt^r
Wt^s )2
(t ^ r
2
Wt^s
)
Nt^s )
(t ^ r
t ^ s))
t ^ s)
2Wt^s (Wt^r
2abWt^s (Wt^r
Wt^s ))
Wt^s ).
Recalling that N, W are martingales, it follows from Theorem 2.57 that M is a martingale
as
abWt^s (Wt^r Wt^s ) = abWs (Wt^r Wt^s ).
Mt =a(Wt^r
=a(Wr
T
In case 2, recalling 0  s  r  u  v, note that
Wt^s )b(Wt^v
Ws )b(Wt^v
Wt^u )
Wt^u )
A
F
as Mt = 0 if t  u. This completes the proof.
Rt
Rt 2
Theorem 3.9. Let f 2 S, Mt = 0 f dW and Nt = Mt2
0 fs ds. Then M and N are
(F⇧ )-martingales. Further, For any T < 1,
Z t
Z T
2
E[sup| f dW | ]  4E[
fs2 ds].
(3.3.6)
tT
0
0
R
Proof. The fact that M and N are martingales follow from Lemma 3.8. As a consequence
E[NT ] = 0 and hence
Z T
Z T
2
E[(
f dW ) ] = E[
fs2 ds].
(3.3.7)
0
0
D
Now the growth inequality (3.3.6) follows from Doob’s maximal inequality (2.24) applied
to N and using (3.3.7).
We will use the growth inequality (3.3.7) to extend the integral to a larger class of
functions that can be approximated in the norm defined by the right hand side in (3.3.7).
e = [0, 1) ⇥ ⌦. It is easy to
Each f 2 S can be viewed as a real valued function on ⌦
e
see that S is an algebra. Let P be the -field generated by S, i.e. the smallest -field on ⌦
such that every element of S is measurable w.r.t. P. Then by Theorem 2.65, the class of
functions that can be approximated by S in the norm
Z T
1
(E[
fs2 ds]) 2
0
78
Chapter 3. The Ito Integral
equals the class of P measurable functions f with finite norm.
The -field P is called the predictable -field. We will discuss the predictable -field
in the next chapter. We note here that every left continuous adapted process X is P
measurable as it is the pointwise limit of
Xtm
= X0 1{0} +
m
m2
X
X
j=0
j
2m
1(
j j+1
,
]
2m 2m
(t).
Lemma 3.10. Let f be a predictable process such that
Z T
E[
fs2 ds] < 1 8T < 1.
(3.3.8)
0
0tT
0
T
Then there exists a continuous adapted process Y such that for all simple predictable processes h 2 S,
Z t
Z T
E[( sup |Yt
hdW |)2 ]  4E[
(fs hs )2 ds] 8T < 1.
(3.3.9)
0
Zt =
Yt2
Z
t
F
Further, Y and Z are (F⇧ )-martingales where
0
fs2 ds.
A
e P) defined as follows: for P measurable
Proof. For r > 0, let µr be the measure on (⌦,
bounded functions g
Z
Z r
gdµr = E[
gs ds]
e
⌦
2
L (µ
0
2
and let us denote the
norm on
r ) by k·k2,µr . By Theorem 2.65, S is dense in L (µr )
for every r > 0 and hence for integers m 1, we can get f m 2 S such that
R
L2
kf
f m k2,µm  2
D
Using k·k2,µr  k·k2,µs for r  s it follows that for k
kf m+k
m 1
f m k2,µm  2
.
(3.3.10)
1
m
.
(3.3.11)
Denoting the L2 (⌦, F, P) norm by k·k2,P , the growth inequality (3.3.6) can be rewritten as,
for g 2 S, m 1,
Z
t
k sup |
0tm
0
gdW | k2,P  2kgk2,µm
(3.3.12)
Rt
Rt
Recall that f k 2 S and hence 0 f k dW is already defined. Let Ytk = 0 f k dW . Now using
(3.3.11) and (3.3.12), we conclude that for k 1
k[ sup |Ytm+k
0tm
Ytm | ]k2,P  2
m+1
.
(3.3.13)
3.3. The Ito Integral
79
Fix an integer n. For m
n, using (3.3.13) for k = 1 we get
k[ sup |Ytm+1
Ytm | ]k2,P  2
0tn
m+1
.
(3.3.14)
and hence
k[
1
X
sup
m=n 0tn
|Ytm+1
Ytm | ]k2,P


1
X
m=n
1
X
k[ sup |Ytm+1
0tn
2
Ytm | ]k2,P
m+1
m=n
<1.
1
X
[ sup |Ytm+1
Ytm |] < 1 a.s. P.
m=n 0tn
Nn = {! :
1
X
[ sup |Ytm+1 (!)
m=n 0tn
Ytm (!)|] = 1}
F
So let
(3.3.15)
T
Hence,
and let N = [1
n=1 Nn . Then N is a P-null set. For ! 62 N , let us define
A
Yt (!) = lim Ytm (!)
m!1
and for ! 2 N , let Yt (!) = 0. It follows from (3.3.15) that for all T < 1, ! 62 N
sup |Ytm (!)
R
0tT
Yt (!)| ! 0.
Thus Y is a process with continuous paths. Now using (3.3.12) for f m
Z t
Z T
m
2
E[( sup |Yt
hdW |) ]  4E[
(f m h)2s ds].
D
0tT
0
(3.3.16)
h 2 S we get
(3.3.17)
0
In view of (3.3.10), the right hand side above converges to
Z T
E[
(fs hs )2 ds].
0
Using Fatou’s lemma and (3.3.16) along with P(N ) = 0, taking lim inf in (3.3.17) we
conclude that (3.3.9) is true. From these observations, it follows that Ytm converges to Yt
in L2 (P) for each fixed t. The observation k·k2,µr  k·k2,µs for r  s and (3.3.10) implies
that for all r, kf f m k2,µr ! 0 and hence for all t
Z t
E[ (fs fsm )2 ds] ! 0.
0
80
Chapter 3. The Ito Integral
As a consequence,
E[
and hence
E[|
Z
Z
t
0
t
0
|(fs )2
(fsm )2 |ds] ! 0
Z
(fsm )2 ds
t
0
fs2 ds| ] ! 0.
Rt n 2
By Theorem 3.9 , we have Y n and Z n are martingales where Ztn = (Ytn )2
0 (fs ) ds.
n
1
n
1
As observed above Yt converges in L (P) to Yt and Zt converges in L (P) to Zt for each
t and hence Y and Z are martingales.
T
Remark 3.11. From the proof of the Lemma it also follows that Y is uniquely determined by
the property (3.3.9), for if Ỹ is another process that satisfies (3.3.9), then using it for h = f m
as in the proof above, we conclude that Y m converges almost surely to Ỹ and hence Y = Ỹ .
F
RT
Definition 3.12. For a predictable process f such that E[ 0 fs2 ds] < 1 8T < 1, we define
Rt
the Ito integral 0 f dW to be the process Y that satisfies (3.3.9).
R
f dW , most of them have
A
The next result gives the basic properties of the Ito integral
essentially been proved above.
Theorem 3.13. Let f, g be predictable processes satisfying (3.3.8).Then
Z t
Z t
Z t
(af + bg)dW = a
f dW + b
gdW.
0
R
0
D
Rt
Let Mt = 0 f dW and Nt = Mt
for any T < 1,
Rt
E[sup|
tT
0
Z
0
(3.3.18)
0
fs2 ds. Then M and N are (F⇧ )-martingales. Further,
t
2
f dW | ]  4E[
Z
T
0
fs2 ds].
(3.3.19)
Proof. The linearity (3.3.18) follows by linearity for the integral for simple functions observed in Lemma 3.7 and then for general predictable processes via approximation. That
M , N are martingales has been observed in Lemma 3.10. The Growth inequality (3.3.19)
follows from (3.3.9).
R
Remark 3.14. For a bounded predictable process f , let IW (f ) = f dW . Then the Growth
inequality (3.3.19) and linearity of IW implies that for fn , f bounded predictable processes
bp
fn ! f implies IW (fn )
ucp
! IW (f ).
3.4. Multidimensional Ito Integral
Exercise 3.15. Let tni = i2
n,
i
81
1 and let
0, n
fsn =
1
X
Wtni 1(tni ,tni+1 ] (s).
i=0
Show that
(i) E[
(ii)
(iii)
RT
Rt
0
Rt
0
0
|fsn
Ws |2 ds] ! 0.
f n dW !
Rt
0
W dW.
W dW = 12 (Wt2
t).
Exercise 3.16. Let ft = t. Show that
Rt
0
f dW = tWt
T
Rt
Hint: For (iii) use 0 f n dW = Btn , (Ant
Btn ) = Qnt along with (Ant + Btn ) = Wt2 (where
An , B n are as in (3.1.9) and (3.1.10)) and Qnt ! t (see Exercise 3.2). Thus Ant and Btn
converge as mentioned in Remark 3.4.
Rt
0
Ws ds.
0, n
1. Observe that
Atni ^t (Wtni+1 ^t
Wtni ^t ) = At Wt
= i2
1
X
i=0
i
R
Hint: Let
n,
A
0
tni
F
Exercise 3.17. Let A 2 V be a bounded r.c.l.l. adapted process with finite variation paths.
Show that
Z t
Z t
A dW = At Wt
W dA.
(3.3.20)
0
1
X
i=0
Wtni+1 ^t (Atni+1 ^t
Atni ^t ).
D
Rt
The left hand side converges to 0 A dW while the second term on right hand side converges
Rt
to 0 W dA as seen in Exercise 2.77.
Remark 3.18. The Ito integral can be extended to a larger class of predictable integrands f
satisfying
Z T
fs2 ds < 1 a.s. 8T < 1.
0
We will outline this later when we discuss integration w.r.t. semimartingales.
3.4
Multidimensional Ito Integral
Let W = (W 1 , W 2 , . . . , W d ) be d-dimensional Brownian motion, where W j is the j th component. In other words, each W j is a real-valued Brownian motion and W 1 , W 2 , . . . , W d
82
Chapter 3. The Ito Integral
are independent. Suppose further that (F⇧ ) is a filtration such that W is (F⇧ ) adapted and
for each s, {Wt Ws : t s} is independent of Fs (and hence (3.2.14) holds). Recall that
defining
d
X
✓
Xt =
✓j Wtj
(3.4.1)
j=1
Mt✓ = (Xt✓ )2
t,
(3.4.2)
8✓ 2 Rd with |✓| = 1; X ✓ , M ✓ are (F⇧ )-martingales.
(3.4.3)
The argument given in the proof of next lemma is interesting and will be used later.
Throughout this section, the filtration (F⇧ ) will remain fixed.
T
Lemma 3.19. For j 6= k, W j W k is also a martingale.
F
Proof. Let Xt = p12 (Wtj + Wtk ). Then, as seen above X is a Brownian motion and hence
Xt2 t is a martingale. Note that
1
t = [(Wtj )2 + (Wtk )2 + 2Wtj Wtk ] t
2
1
1
= [(Wtj )2 t] + [(Wtk )2 t] + Wtj Wtk .
2
2
Since the left hand side above as well as the first two terms of right hand side above are
martingales, it follows that so is the third term.
R
A
Xt2
D
Suppose that for 1  j  m and 1  k  d, f jk are (F⇧ )-predictable processes that
satisfy (3.3.8). Let
d Z t
X
Xtj =
f jk dW k .
k=1
0
Let X = (X 1 , . . . X m ) denote the m-dimensional process. It is natural to define Xt to be
Rt
predictable process. We
0 f dX where we interpret f to be L(m, d) (m ⇥ d-matrix-valued)
Rt
will obtain a growth estimate on the stochastic integral 0 f dW which in turn would be
crucial in the study of stochastic di↵erential equations. The following lemma is a first step
towards it.
Lemma 3.20. Let h be a predictable process satisfying (3.3.8). Let Ytk =
for j 6= k, Ytk Ytj is a martingale.
Rt
0
hdW k . Then
3.4. Multidimensional Ito Integral
83
T
Proof. Let Xt = p12 (Wtj + Wtk ). Then, as seen above X is a Brownian motion. For simple
functions f it is easy to check that
Z t
Z t
Z t
1
f dX = p ( f dW j +
f dW k )
2 0
0
0
and hence for all f satisfying (3.3.8) via approximation. Thus,
Z t
1
hdX = p (Ytj + Ytk )
2
0
Rt
R
t 2
and so ( 0 hdX)2
0 hs ds is a martingale. Now as in the proof of Lemma 3.19
Z t
Z t
Z t
1
j 2
j k
2
2
k 2
( hdX)
hs ds = [(Yt ) + (Yt ) + 2Yt Yt ]
h2s ds
2
0
0
0
Z t
Z t
1
1
= [(Ytj )2
h2s ds] + [(Ytk )2
h2s ds] + Ytj Ytk .
2
2
0
0
F
and once again the left hand side as well as the first two terms on the right hand side are
martingales and hence so is the last term completing the proof.
Rt
Lemma 3.21. Let f, g be predictable processes satisfying (3.3.8). Let Ytk = 0 f dW k ,
Rt
Ztk = 0 gdW k . Then for j 6= k, Ytk Ztj is a martingale.
R
A
Proof. Let Xt = Ytk Ztj . We will first prove that X is a martingale when f, g are simple, the
general case follows by approximation. The argument is similar to the proof of Theorem
3.8. By linearity, suffices to prove the required result in the following cases.
Case 1: 0  s < r and
ft = a1(s,r] (t), gt = b1(s,r] (t), a, b are Fs measurable, a
0, b
0.
D
Case 2: 0  s < r  u < v and
ft = a1(s,r] (t), gt = b1(u,v] (t), a is Fs measurable and b is Fu measurable.
In Case 1,
k
Xt = Ytk Ztj = ab(Wr^t
j
k
Ws^t
)(Wr^t
j
Ws^t
)=(
Z
t
0
hdW k )(
Z
p
where h = (ab)1(s,r] (t) and hence by Lemma 3.20, X is a martingale.
In Case 2,
j
j
k
k
Xt = Ytk Ztj = ab(Wr^t
Ws^t
)(Wv^t
Wu^t
).
Here, Xt = 0 for t  a and
j
Xt = ⇠(Wv^t
j
Wu^t
)
t
0
hdW j )
84
Chapter 3. The Ito Integral
k
with ⇠ = ab(Wr^t
k ) is F measurable and hence by Theorem 2.57, X is a martingale.
Ws^t
u
This proves the required result for simple predictable processes f, g. The general case
follows by approximating f, g by simple predictable processes {f n }, {g n } such that for all
T <1
Z
T
0
[|fsn
fs |2 + |gsn
gs |2 ]ds ! 0.
Then it follows from (3.3.19) that for each t < 1,
Z t
Z t
n
k
f dW !
f dW k in L2 (P),
0
Z
0
t
0
g n dW j !
Z
t
0
gdW j in L2 (P)
T
Rt
Rt
and hence the martingale ( 0 f n dW k )( 0 g n dW j ) converges to
Rt
Rt
Ytk Ztj = ( 0 f dW k )( 0 gdW j ) in L1 (P) and thus Ytk Ztj is a martingale.
A
F
We are now ready to prove the main growth inequality. Recall that
L(d, m) denotes
qP
d
d
2
the space of d ⇥ m matrices and for x = (x1 , x2 , . . . , xd ) 2 R , |x| =
j=1 xj denotes
qP
Pm 2
d
the Euclidean norm on Rm and for a = (ajk ) 2 L(d, m), kak =
j=1
k=1 ajk is the
Euclidean norm on L(d, m).
R
Let f = (f jk ) be L(d, m)-valued process. For Rd -valued Brownian motion W , we have
R
seen that X = f dW is an Rm -valued process.
we have
D
Theorem 3.22. Let W be an Rd -valued Brownian motion. Then for m ⇥ d-matrix-valued
predictable process f with
Z T
E[
kfs k2 ds] < 1
0
E[ sup |
0tT
Proof. Let Xtj =
Pd
(Xtj )2
k=1
R
t
0
2
f dW | ]  4E[
Z
T
0
kfs k2 ds].
(3.4.4)
f jk dW k . Then
d Z
X
k=1
Z
t
0
Z t
d
X
(f ) ds =
[( f jk dW k )2
Z
jk 2
k=1
+
0
d X
d
X
l=1 k=1
1{k6=l}
Z
t
t
(f jk )2 ds]
0
jk
f dW
0
k
Z
t
0
f jl dW l
3.5. Stochastic Di↵erential Equations
85
and each term on the right hand side above is a martingale and thus summing over j we
conclude
Z t
Z t
| f dW |2
kfs k2 ds
0
is a martingale. Thus
E[|
Z
t
0
0
f dW |2 ] = E[
Z
t
0
kfs k2 ds]
(3.4.5)
Rt
and | 0 f dW |2 is a submartingale. The required estimate (3.4.4) now follows from Doob’s
maximal inequality (2.24) and (3.4.5).
3.5
Stochastic Di↵erential Equations
T
We are going to consider stochastic di↵erential equations indexstochastic di↵erential equations (SDE) of the type
dXt = (t, Xt )dWt + b(t, Xt )dt.
(3.5.1)
F
The equation (3.5.1) is to be interpreted as an integral equation :
Z t
Z t
Xt = X0 +
(s, Xs )dWs +
b(s, Xs )ds.
0
(3.5.2)
0
R
A
Here W is an Rd -valued Brownian motion, X0 is a F0 measurable random variable, :
[0, 1) ⇥ Rm 7! L(m, d) and b : [0, 1) ⇥ Rm 7! Rm are given functions and one is seeking a
process X such that (3.5.2) is true. The solution X to the SDE (3.5.1), when it exists, is
⇤ and drift coefficient b.
called a di↵usion process with di↵usion coefficient
We are going to impose the following conditions on , b:
: [0, 1) ⇥ Rm 7! L(m, d) is a continuous function
(3.5.3)
D
b : [0, 1) ⇥ Rm 7! Rm is a continuous function
8T < 1 9CT < 1 such that for all t 2 [0, T ], x1 , x2 2 Rd
k (t, x1 )
1
|b(t, x )
(t, x2 )k  CT |x1
x2 |,
b(t, x )|  CT |x
x |.
2
1
(3.5.4)
2
Since t 7! (t, 0) and t 7! b(t, 0) are continuous and hence bounded on [0, T ] for every
T < 1, using the Lipschitz conditions (3.5.4), we can conclude that for each T < 1,
9KT < 1 such that
k (t, x)k  KT (1 + |x|),
|b(t, x)|  KT (1 + |x|).
(3.5.5)
86
Chapter 3. The Ito Integral
We will need the following lemma, known as Gronwall’s lemma for proving uniqueness of
solution to (3.5.2) under the Lipschitz conditions.
Lemma 3.23. Let (t) be a bounded measurable function on [0, T ] satisfying, for some
0  a < 1, 0 < b < 1,
Z t
(t)  a + b
(s) ds, 0  t  T.
(3.5.6)
0
Then
(t)  aebt .
g(t) = e
bt
Z
t
(s) ds.
T
Proof. Let
(3.5.7)
0
t
(s) ds a.e.
F
Then by definition, g is absolutely continuous and
Z
0
bt
bt
g (t) = e
(t) be
0
A
where almost everywhere refers to the Lebesgue measure on R. Using (3.5.6), it follows
that
g 0 (t)  ae
bt
a.e.
e
bt )
from which
R
Hence (using g(0) = 0 and that g is absolutely continuous) g(t)  ab (1
we get
Z t
a
(s) ds  (ebt 1).
b
0
D
The conclusion (t)  aebt follows immediately from (3.5.6).
So now let (F⇧ ) be a filtration on (⌦, F, P) and W be a d-dimensional Brownian motion
adapted to (F⇧ ) and such that for s < t, Wt Ws is independent of Fs . Without loss of
generality, let us assume that (⌦, F, P) is complete and that F0 contains all P null sets in
F.
Let Km denote the class of Rm -valued continuous (F· ) adapted process Z such that
RT
E[ 0 |Zs |2 ds] < 1 8T < 1. For Y 2 Km let
Z t
Z t
⇠t = Y0 +
(s, Ys )dWs +
b(s, Ys )ds.
(3.5.8)
0
0
3.5. Stochastic Di↵erential Equations
87
Note that in view of the growth condition (3.5.5) the Ito integral above is defined. Using
the growth estimate (3.4.4) we see that
Z T
2
2
E[ sup |⇠t | ] =3[E[|Y0 | ] + 4E[
k (s, Ys )k2 ds]
0tT
+ E[(
Z
0
T
0
|b(s, Ys )|ds)2 ]]
2
3E[|Y0 | ] +
3KT2 (4
+ T)
Z
T
0
(1 + E[|Ys |2 ])ds
T
and hence ⇠ 2 Km . Let us define a mapping ⇤ from Km into itself as follows: ⇤(Y ) = ⇠
where ⇠ is defined by (3.5.8). Thus solving the SDE (3.5.2) amounts to finding a fixed
point Z of the functional ⇤ with Z0 = X0 , where X0 is pre-specified. We are going to
prove that given X0 , there exists a unique solution (or a unique fixed point of ⇤) with the
given initial condition. The following lemma is an important step in that direction.
F
Lemma 3.24. Let Y, Z 2 Km and let ⇠ = ⇤(Y ) and ⌘ = ⇤(Z). Then for 0  t  T one
has
Z t
2
2
2
E[ sup |⇠s ⌘s | ]  3E[|Y0 Z0 | ] + 3CT (4 + T )
E[|Ys Zs |2 ]ds
Proof. Let us note that
⇠t
⌘t = Y0
Z0 +
Z
t
A
0st
[ (s, Ys )
0
(s, Zs )]dWs +
Z
0
t
[b(s, Ys )
b(s, Zs )]ds
0
R
and hence this time using the Lipschitz condition (3.5.4) along with the growth inequality
(3.4.4) we now have
Z t
E[ sup |⇠s ⌘s |2 ] 3[E[|Y0 Z0 |2 ] + 4E[ k (s, Ys )
(s, Zs )k2 ds]
D
0st
0
t
b(s, Zs )|ds)2 ]]
0
Z t
2
2
Z0 | ] + 3CT (4 + T )
E[|Ys Zs |2 ]ds.
+ E[(
3E[|Y0
Z
|b(s, Ys )
0
Corollary 3.25. Suppose Y, Z 2 Km be such that Y0 = Z0 . Then for 0  t  T
Z t
2
2
E[ sup |⇤(Y )s ⇤(Z)s | ]  3CT (4 + T )
E[|Ys Zs |2 ]ds
0st
0
We are now in a position to prove the main result of this section.
88
Chapter 3. The Ito Integral
Theorem 3.26. Suppose , b satisfy conditions (3.5.3) and (3.5.4) and X0 is a F0 measurable Rm -valued random variable with E[|X0 |2 ] < 1. Then there exists a process X such
RT
that E[ 0 |Xs |2 ds] < 1 8T < 1 and
Z t
Z t
Xt = X0 +
(s, Xs )dWs +
b(s, Xs )ds.
(3.5.9)
0
0
RT
Further if X̃ is another process such that X̃0 = X0 , E[ 0 |X̃s |2 ds] < 1 for all T < 1 and
Z t
Z t
X̃t = X̃0 +
(s, X̃s )dWs +
b(s, X̃s )ds
0
0
then X = X̃, i.e. P(Xt = Yt 8t) = 1.
u(t) = E[ sup|Xs
st
u(t)  3CT2 (4 + T )
u(t) 
t
Z
X̃s |2 ]ds.
E[|Xs
0
t
A
Hence u is bounded and satisfies
Z
X̃s |2 ]
F
satisfies for 0  t  T (recalling X0 = X̃0 )
T
Proof. Let us first prove uniqueness. Let X and X̃ be as in the statement of the theorem.
Then, using Lemma (3.24) it follows that
3CT2 (4
+ T)
0
u(s)ds, 0  t  T.
D
R
By (Gronwall’s) Lemma (3.23), it follows that u(t) = 0, 0  t  T for every T < 1. Hence
X = X̃.
We will now construct a solution. Let Xt1 = X0 for all t
0. Note that X 1 2 Km .
Now define X n inductively by
X n+1 = ⇤(X n ).
Since Xt1 = X0 for all t and X02 = X01 ,
Z t
Z t
2
1
Xt Xt =
(s, X0 )dWs +
b(s, X0 )ds
0
0
and hence
E[sup|Xs2
st
Xs1 |2 ]  2KT2 (4 + T )(1 + E[|X0 |2 ])t.
Note that X0n = X01 = X0 for all n
n 2, for 0  t  T ,
E[sup|Xsn+1
st
(3.5.10)
1 and hence from Lemma (3.24) it follows that for
Xsn |2 ]  3CT2 (4 + T )
Z
t
0
E[|Xsn
Xsn
1 2
| ]ds
3.5. Stochastic Di↵erential Equations
Thus defining for n
89
1, un = E[supst |Xsn+1
Xsn |2 ] we have for n
Z t
2
un (t)  3CT (4 + T )
un 1 (s)ds.
2, for 0  t  T ,
(3.5.11)
0
As seen in (3.5.10),
u1 (t)  2KT2 (4 + T )(1 + E[|X0 |2 ])t
and hence using (3.5.11), which is true for n 2, we can deduce by induction on n that
for a constant C̃T = 3(CT2 + KT2 )(4 + T )(1 + E[|X0 |2 ])
P p
n
(C̃T )n tn
, 0  t  T.
n!
un (T ) < 1 for every T < 1 which is same as
1
X
ksup|Xsn+1
n=1 sT
T
Thus
un (t) 
Xsn |k2 < 1
(3.5.12)
1
X
as well as
D
sT
Xsn | ]k2 < 1
Xsn | ]k2  supk[
R
supk[sup|Xsn+k
k 1
sup|Xsn+1
n=1 sT
A
k[
F
kZk2 denoting the L2 (P) norm here. The relation (3.5.12) implies
k 1
[
1
X
n+k
X
(3.5.13)
sup|Xsj+1
j=n sT
ksup|Xsj+1
j=n sT
Xsj | ]k2
Xsj |k2 ]
(3.5.14)
! 0 as n tends to 1.
P1
n+1 (!)
Let N = [1
Xsn (!)| = 1}. Then by (3.5.13) P(N ) = 0
n=1 supsT |Xs
T =1 {! :
and for ! 62 N , Xsn (!) converges uniformly on [0, T ] for every T < 1. So let us define X
as follows:
8
<lim
n
c
n!1 Xt (!) if ! 2 N
Xt (!) =
:0
if ! 2 N.
By definition, X is a continuous adapted process (since by assumption N 2 F0 ) and X n
converges to X uniformly in [0, T ] for every T almost surely. Using Fatou’s lemma and
90
Chapter 3. The Ito Integral
(3.5.14) we get
k[sup|Xs
sT
Xsn | ]k2  lim inf k[
k!1
[
1
X
n+k
X
sup|Xsj+1
j=n sT
ksup|Xsj+1
j=n sT
Xsj | ]k2
Xsj |k2 ]
(3.5.15)
! 0 as n tends to 1.
In particular, X 2 Km . Since ⇤(X n ) = X n+1 by definition, (3.5.15) also implies that
⇤(X n )s | ]k2 = 0
lim k[sup|Xs
n!1
sT
(3.5.16)
while (3.5.15) and Corollary 3.25 (remembering that X0n = X0 for all n) imply that
sT
⇤(X n )s | ]k2 = 0.
T
lim k[sup|⇤(X)s
n!1
(3.5.17)
D
R
A
F
From (3.5.16) and (3.5.17) it follows that X = ⇤(X) or that X is a solution to the SDE
(3.5.9).
Chapter 4
Stochastic Integration
0
T
In this chapter we will pose the question as to what processes X are good integrators: i.e.
Z t
JX (f )(t) =
f dX
R
A
F
can be defined for a suitable class of integrands f and has some natural continuity properties. We will call such a process a Stochastic integrator. In this chapter, we will prove
Rt
basic properties of the stochastic integral 0 f dX for a stochastic integrator X.
In the rest of the book, (⌦, F, P) will denote a complete probability space and (F⇧ ) will
denote a filtration such that F0 contains all null sets in F. All notions such as adapted,
stop time, martingale will refer to this filtration unless otherwise stated explicitly.
For some of the auxiliary results, we need to consider the corresponding right continuous
filtration (F⇧+ ) = {Ft+ : t 0} where
Ft+ = \s>t Fs .
4.1
D
We begin with a discussion on the predictable -field.
The Predictable -Field
e = [0, 1) ⇥ ⌦
Recall our convention that a process X = (Xt ) is viewed as a function on ⌦
e generated by S. Here S
and the predictable -field P has been defined as the -field on ⌦
consists of simple adapted processes:
m
X
f (s) = a0 1{0} (s) +
ak+1 1(sk ,sk+1 ] (s)
(4.1.1)
k=0
where 0 = s0 < s1 < s2 < . . . < sm+1 < 1, ak is bounded Fsk 1 measurable random
variable, 1  k  (m + 1) and a0 is bounded F0 measurable. P measurable processes have
91
92
Chapter 4. Stochastic Integration
appeared naturally in the definition of the stochastic integral w.r.t. Brownian motion and
play a very significant role in the theory of stochastic integration with respect to general
semimartingales as we will see. A process f will be called a predictable process if it is P
measurable. Of course, P depends upon the underlying filtration and would refer to the
filtration that we have fixed. If there are more than one filtration under consideration, we
will state it explicitly. For example P(G· ) to denote the predictable -field corresponding
to a filtration (G· ) and S(G· ) to denote simple predictable process for the filtration (G· ).
The following proposition lists various facts about the -field P.
Proposition 4.1. Let (F⇧ ) be a filtration and P = P(F· ).
T
(i) Let f be P measurable. Then f is (F⇧ ) adapted. Moreover, for every t < 1, ft is
([s<t Fs )-measurable.
(ii) Let Y be a left continuous adapted process. Then Y is P measurable.
F
(iii) Let T be the class of all bounded adapted continuous processes. Then P = (T) and
e P).
the smallest bp- closed class that contains T is B(⌦,
A
(iv) For any stop time ⌧ , U = 1[0,⌧ ] (i.e. Ut = 1[0,⌧ ] (t)) is P measurable.
(v) For an r.c.l.l. adapted process Z and a stop time ⌧ , the process X defined by
is predictable.
(4.1.2)
R
Xt = Z⌧ 1(⌧,1) (t)
D
(vi) For a predictable process g and a stop time ⌧ , g⌧ is a random variable and h defined
by
ht = g⌧ 1(⌧,1) (t)
(4.1.3)
is itself predictable.
Proof. It suffices to prove the assertions assuming that the processes f , Y , g are bounded
(by making a tan 1 transformation, if necessary). Now, for (i) let
e P) : ft is ([s<t Fs )- measurable}.
K1 = {f 2 B(⌦,
It is easily seen that K1 is bp-closed and contains S and thus by Theorem 2.64 equals
e P) proving (i).
B(⌦,
4.1. The Predictable -Field
93
For (ii), given a left continuous bounded adapted process Y , let Y n be defined by
n
Ytn
= Y0 1{0} (t) +
bp
n2
X
Y
k=0
k
2n
1( kn , k+1
(t).
n ]
2
(4.1.4)
2
Then Y n 2 S and Y n ! Y and this proves (ii).
For (iii), Let K2 be the smallest bp-closed class containing T. From part (ii) above, it
e P) and hence K2 ✓ B(⌦,
e P). For t 2 [0, 1), n 1 let
follows that T ✓ B(⌦,
n (t)
= (1
nt)1[0, 1 ] (t),
n
n (t) = nt1(0, 1 ] (t) + 1( 1 ,1] (t) + (1
n
n
n(t
1))1(1,1+ 1 ] (t).
n
n bp
!
j=0
1{0} and
n bp
!
1(0,1] .
T
Then n and n are continuous functions, bounded by 1 and
For f 2 S given by
m
X
f (s) = a0 1{0} (s) +
aj+1 1(sj ,sj+1 ] (s)
F
where 0 = s0 < s1 < s2 . . . < sm+1 < 1, aj+1 is bounded Fsj measurable random variable,
0  j  m, and a0 is bounded F0 measurable random variable let
m
X
s s
n
n
Ys = a0 (s) +
aj+1 n ( sj+1 jsj ).
A
j=0
bp
D
R
e P) ✓ K2
Then it follows that Y n 2 T and Y n ! Y . Thus S ✓ K2 and hence B(⌦,
completing the proof of (iii).
For part (iv) note that U is adapted left continuous process and hence P measurable
by part (ii).
For (v), suffices to prove that X is adapted since X is left continuous by construction.
We have shown earlier (Lemma 2.36) that Z⌧ is F⌧ measurable and hence Da,t = {Z⌧ 
a} \ {⌧  t} belongs to Ft . Note that for a < 0
while for a
0,
{Xt  a} = {Z⌧  a} \ {⌧ < t}
= Da,t \ {⌧ < t}
{Xt  a} = (Da,t \ {⌧ < t}) [ {⌧
t}
and hence {Xt  a} 2 Ft for all a. This proves (v).
For (vi), the class of processes g for which (vi) is true is closed under bp convergence
and contains the class of continuous adapted processes as shown above. In view of part
(iii), this completes the proof.
94
Chapter 4. Stochastic Integration
If X is P(F⇧Y ) measurable, then part (i) of the result proven above says that for every
t, Xt is measurable w.r.t. (Yu : 0  u < t). Thus having observed Yu , u < t, the value Xt
can be known (predicted with certainty) even before observing Yt . This justifies the name
predictable -field for P.
Exercise 4.2. For t
0, let Gt = Ft+ . Show that for t > 0
([s<t Fs ) = ([s<t Gs ).
Here is an important observation regarding the predictable -field P(F⇧+ ) corresponding
to the filtration (F⇧+ ).
T
Theorem 4.3. Let f be a (F⇧+ )-predictable process (i.e. P(F⇧+ ) measurable). Then g
defined by
gt (!) = ft (!)1{(0,1)⇥⌦} (t, !)
(4.1.5)
is a (F⇧ )-predictable process.
F
Proof. Let f a (F⇧+ )-adapted bounded continuous process. A crucial observation is that
then for t > 0, ft is Ft measurable and thus defining for n 1
hnt = (nt ^ 1)ft
bp
D
R
A
it follows thhat hn are (F⇧ ) adapted continuous processes and hn ! g where g is defined
by (4.1.5) and hence in this case g is (F⇧ ) predictable.
Now let H be the class of bounded P(F⇧+ ) measurable f for which the conclusion is
true. Then easy to see that H is an algebra that is bp-closed and contains all (F⇧+ )-adapted
continuous processes. The conclusion follows by the monotone class theorem, Theorem
2.64.
The following is an immediate consequence of this.
Corollary 4.4. Let f be a process (F⇧+ )-predictable such that f0 is F0 measurable. Then
f is (F⇧ )-predictable.
Proof. Since f0 is F0 measurable, ht = f0 1{0} (t) is (F⇧ )-predictable and f = g + h where g
is the (F⇧ )-predictable process given by (4.1.5). Hence f is (F⇧ )-predictable.
Corollary 4.5. Let A ✓ (0, 1) ⇥ ⌦. Then A 2 P(F⇧ ) if and only if A 2 P(F⇧+ ).
The following example will show that the result may not be true if f0 = 0 is dropped
in the Corollary 4.4 above.
4.2. Stochastic Integrators
95
Exercise 4.6. Let ⌦ = C([0, 1) and let Xt denote the coordinate process and Ft = (Xs :
0  s  t). Let A be the set of all ! 2 ⌦ that take positive as well as negative values in
(0, ✏) for every ✏ > 0. Show that A 2 F0+ but A does not belong to F0 . Use this to show the
relevance of the hypothesis on f0 in the Corollary given above.
Exercise 4.7. Let ⇠ be an F⌧ measurable random variable. Show that Y = ⇠ 1(⌧,1) is
predictable.
Hint: Use Lemma 2.39 along with part (v) in Proposition 4.1.
4.2
Stochastic Integrators
j=0
T
Let us fix an r.c.l.l. (F⇧ ) adapted stochastic process X.
Recall, S consists of the class of processes f of the form
m
X
f (s) = a0 1{0} (s) +
aj+1 1(sj ,sj+1 ] (s)
(4.2.1)
A
F
where 0 = s0 < s1 < s2 < . . . < sm+1 < 1, aj is bounded Fsj 1 measurable random
variable, 1  j  (m + 1), and a0 is bounded F0 measurable.
For a simple predictable f 2 S given by (4.2.1), let JX (f ) be the r.c.l.l. process defined
by
m
X
JX (f )(t) = a0 X0 +
aj+1 (Xsj+1 ^t Xsj ^t ).
(4.2.2)
j=0
D
R
One needs to verify that JX is unambiguously defined on S. i.e. if a given f has two
representations of type (4.2.1), then the corresponding expression in (4.2.2) agree. This as
well as linearity of JX (f ) for f 2 S can be verified using elementary algebra. By definition,
for f 2 S, JX (f ) is an r.c.l.l. adapted process. In analogy with the Ito integral with respect
to Brownian motion discussed in the earlier chapter, we wish to explore if we can extend JX
to the smallest bp-closed class of integrands that contain S. Each f 2 S can be viewed as a
e = [0, 1) ⇥ ⌦. Since P is the -field generated by S, by Theorem
real valued function on ⌦
2.64, the smallest class of functions that contains S and is closed under bp-convergence is
e P).
B(⌦,
When the space, filtration and the probability measure are clear from the context, we
will write the class of adapted r.c.l.l. processes R0 (⌦, (F⇧ ), P) simply as R0 .
Definition 4.8. An r.c.l.l. adapted process X is said to be a Stochastic integrator if the mape P) 7! R0 (⌦, (F⇧ ), P) satisfying
ping JX from S to R0 (⌦, (F⇧ ), P) has an extension JX : B(⌦,
96
Chapter 4. Stochastic Integration
the following continuity property:
bp
f n ! f implies JX (f n )
ucp
! JX (f ).
(4.2.3)
It should be noted that for a given r.c.l.l. process X, JX may not be continuous on S.
See the next Exercise. So this definition , in particular, requires that JX is continuous on
e P). Though not easy to prove, continuity of JX
S and has a continuous extension to B(⌦,
on S does imply that it has a continuous extension and hence it is a stochastic integrator.
We will see this later in Section 5.8.
m
m
0 = tm
0 < t1 < . . . < tnm = T
(4.2.4)
F
T
Exercise 4.9. Let G be a real valued function from [0, 1) with G(0) = 0 that does not
have bounded variation. So for some T < 1, Var[0,T ] (G) = 1. We will show that JG is not
continuous on S. From the definition of Var[0,T ] (G), it follows that we can get a sequence of
partitions of [0, T ]
(4.2.5)
such that
↵m = (
nX
m 1
j=0
|G(tm
j )
G(tm
j 1 )| ) ! 1.
A
Let sgn : R !
7 R be defined by sgn(x) = 1 for x
0 and sgn(x) =
1
m
m
m
m . Show that
f = (↵m ) 2 sgn(G(tj ) G(tj 1 ))1(tm
j 1 ,tj ]
1 for x < 0. Let
R
(i) f m converges to 0 uniformly.
(ii) JG (f m ) converges to 1.
D
Conclude that JG is not continuous on S.
We next observe that the extension, when it exists is unique.
0 from
Theorem 4.10. Let X be an r.c.l.l. process. Suppose there exist mappings JX , JX
e P)into R0 (⌦, (F⇧ ), P), such that for f 2 S (given by (4.2.1)),
B(⌦,
0
JX (f )(t) = JX
(f )(t) = a0 X0 +
m
X
j=1
aj+1 (Xsj+1 ^t
Xsj ^t ).
0 satisfy (4.2.3). Then
Further suppose that both JX , JX
0
e P).
(f )(t) 8t] = 1 8f 2 B(⌦,
P[JX (f )(t) = JX
(4.2.6)
4.2. Stochastic Integrators
97
0 both satisfy (4.2.3), the class
Proof. Since JX , JX
e P) : P(JX (f )(t) = J 0 (f )(t) 8t) = 1}
K1 = {f 2 B(⌦,
X
is bp-closed and by our assumption (4.2.6), contains S. Since P = (S), by Theorem 2.64
e P).
it follows that K1 = B(⌦,
The following result which is almost obvious in this treatment of stochastic integration
is a deep result in the traditional approach to stochastic integration and is known as
Stricker’s theorem.
T
Theorem 4.11. Let X be a stochastic integrator for the filtration (F⇧ ). Let (G⇧ ) be a
filtration such that Ft ✓ Gt for all t. Suppose X is a stochastic integrator for the filtration
(G⇧ ) as well. Denoting the mapping defined by (4.2.2) for the filtration (G⇧ ) and its extension
by HX , we have
e P(F⇧ )).
JX (f ) = HX (f ) 8f 2 B(⌦,
(4.2.7)
F
0 be the restriction of H
0
e
Proof. Let JX
X to B(⌦, P(F⇧ )). Then JX satisfies (4.2.6) as well
as (4.2.3). Thus (4.2.7) follows from uniqueness of extension, Theorem 4.10.
A
Here is an observation that plays an important role in next result.
R
Lemma 4.12. Let X be a stochastic integrator and ⇠ be a F0 measurable bounded random
e P)
variable. Then 8f 2 B(⌦,
JX (⇠f ) = ⇠JX (f ).
(4.2.8)
D
e P) such that (4.2.8) is true. Easy to verify that
Proof. Let K2 consist of all f 2 B(⌦,
e P) by Theorem 2.64.
S ✓ K2 and that K2 is bp-closed. Thus, K2 = B(⌦,
The next observation is about the role of P-null sets.
e P),
Theorem 4.13. Let X be a stochastic integrator. Then f, g 2 B(⌦,
P(! 2 ⌦ : ft (!) = gt (!) 8t
0) = 1
(4.2.9)
implies
P(! 2 ⌦ : JX (f )t (!) = JX (g)t (!) 8t
0) = 1.
(4.2.10)
In other words, the mapping JX maps equivalence classes of process under the relation
f = g (see Definition (2.2)) to equivalence class of processes.
98
Chapter 4. Stochastic Integration
Proof. Given f, g such that (4.2.9) holds, let
⌦0 = {! 2 ⌦ : ft (!) = gt (!) 8t
0}.
Then the assumption that F0 contains all P-null sets implies that ⌦0 2 F0 and thus ⇠ = 1⌦0
is F0 measurable and ⇠f = ⇠g in the sense that these are identical processes:
⇠(!)ft (!) = ⇠(!)gt (!) 8! 2 ⌦, t
0.
Now we have
⇠JX (f ) = JX (⇠f )
= JX (⇠g)
= ⇠JX (g).
T
Since P(⇠ = 1) = 1, (4.2.10) follows.
F
Corollary 4.14. Let fn , f be bounded predictable be such that there exists ⌦0 ✓ ⌦ with
P(⌦0 ) = 1 and such that
bp
1⌦0 fn ! 1⌦0 f.
Then
ucp
A
JX (fn )
! JX (f ).
R
Remark 4.15. Equivalent Probability Measures : Let Q be a probability measure equivalent
to P. In other words, for A 2 F,
Q(A) = 0 if and only if P(A) = 0.
D
Then it is well known and easy to see that (for a sequence of random variables) convergence
in P probability implies and is implied by convergence in Q probability. Thus, it follows that an
r.c.l.l. adapted process X is a stochastic integrator on (⌦, F, P) if and only if it is a stochastic
integrator on (⌦, F, Q). Moreover, the class L(X) under the two measures is the same and for
R
f 2 L(X), the stochastic integral f dX on the two spaces is identical.
We have thus seen that the stochastic integrator does not quite depend upon the underlying probability measure. We have also seen that it does not depend upon the underlying
filtration either - see Theorem 4.11.
R
For a stochastic integrator X, we will be defining the stochastic integral Y = f dX,
for an appropriate class of integrands, given as follows:
4.2. Stochastic Integrators
99
Definition 4.16. For a stochastic integrator X, let L(X) denote the class of predictable
processes f such that
e P), hn ! 0 pointwise, |hn |  |f | ) JX (hn )
hn 2 B(⌦,
ucp
! 0.
(4.2.11)
From the definition, it follows that if f 2 L(X) and g is predictable such that |g|  |f |,
then g 2 L(X). Here is an interesting consequence of this definition.
Theorem 4.17. Let X be a stochastic integrator. Then f 2 L(X) if and only if
e P), g n ! g pointwise , |g n |  |f | ) JX (g n ) is Cauchy in ducp .
g n 2 B(⌦,
(4.2.12)
Proof. Suppose f satisfies (4.2.11). Let
T
e P), g n ! g pointwise , |g n |  |f |.
g n 2 B(⌦,
Given any subsequences {mk }, {nk } of integers, increasing to 1, let
1
k
hk = (g m
2
k
(JX (g m )
ucp
F
Then {hk } satisfies (4.2.11) and hence JX (hk )
k
g n ).
! 0 and as a consequence
k
JX (g n ))
ucp
! 0.
(4.2.13)
D
R
A
Since (4.2.13) holds for all subsequences {mk }, {nk } of integers, increasing to 1, it follows
that JX (g n ) is Cauchy. Conversely, suppose f satisfies (4.2.12). Given {hn } as in (4.2.11),
let a sequence g k be defined as g 2k = hk and g 2k 1 = 0 for k
1. Then g k converges
to g = 0 and hence JX (g k ) is Cauchy. Since for odd integers n, JX (g n ) = 0 and thus
JX (g 2k ) = JX (hk ) converges to 0.
R
This result enables us to define f dX for f 2 L(X).
Definition 4.18. Let X be a stochastic integrator and let f 2 L(X). Then
Z
f dX = lim JX (f 1{|f |n} )
n!1
(4.2.14)
where the limit is in the ducp metric.
Exercise 4.19. Let an ! 1. Show that for f 2 L(X),
Z
ucp
JX (f 1{|f |an } ) ! f dX.
R
When f is bounded, it follows that f dX = JX (f ) by definition. Here is an important
result which is essentially a version of dominated convergence theorem.
100
Chapter 4. Stochastic Integration
e P) be such that g n ! g pointwise, |g n |  |f |.
Theorem 4.20. Let f 2 L(X) and g n 2 B(⌦,
Then
Z
Z
n
g dX ! gdX in ducp metric as n ! 1.
Proof. For n 1, let ⇠ 2n 1 = g 1{|g|n} and ⇠ 2n = g n . Then |⇠ m |  |f | and ⇠ m converges
R
pointwise to g. Thus, ⇠ m dX is Cauchy in ducp metric. On the other hand
Z
Z
ucp
2n 1
⇠
dX ! gdX
R
from the definition of gdX. Thus
Z
Z
Z
ucp
2n
n
⇠ dX = g dX ! gdX.
F
T
Note that in the result given above, we did not require g to be bounded. Even if g were
bounded, the convergence was not required to be bounded pointwise.
R
The process f dX is called the stochastic integral of f with respect to X and we will
also write
Z
Z
t
(
f dX)t =
f dX.
0
A
Rt
We interpret 0 f dX as the definite integral of f with respect to X over the interval [0, t].
We sometime need the integral of f w.r.t. X over (0, t] and so we introduce
Z t
Z t
Z t
f dX =
f 1(0,1) dX =
f dX f0 X0 .
0+
0
0
R
Rt
Note that f dX is an r.c.l.l. process by definition. We will also write 0 f dX to denote
Rs
Yt where Ys = 0 f dX.
A simple example of a Stochastic Integrator is an r.c.l.l. adapted process X such that
the mapping t 7! Xt (!) satisfies Var[0,T ] (X· (!)) < 1 for all ! 2 ⌦, for all T < 1.
D
R
Theorem 4.21. Let X 2 V be a process with finite variation paths, i.e. X be an r.c.l.l.
adapted process such that
Var[0,T ] (X· (!)) < 1 for all T < 1.
(4.2.15)
e P) the stochastic integral JX (f ) =
Then X is a stochastic integrator. Further, for f 2 B(⌦,
R
f dX is the Lebesgue-Stieltjes integral for every ! 2 ⌦:
Z t
JX (f )(t)(!) =
f (s, !)dXs (!)
(4.2.16)
0
where the integral above is the Lebesgue-Stieltjes integral.
4.3. Properties of the Stochastic Integral
101
Proof. For f 2 S, the right hand side of (4.2.16) agrees with the specification in (4.2.1)(4.2.2). Further the dominated convergence theorem (for Lebesgue-Stieltjes integral) implies that the right hand side of (4.2.16) satisfies (4.2.3) and thus X is a stochastic integrator
and (4.2.16) is true.
As seen in Remark 3.14, Brownian motion W is also a stochastic integrator.
Remark 4.22. If X 2 V and At = |X|t is the total variation of X on [0, t] and f is predictable
such that
Z
t
0
|fs |dAs < 1 8t < 1 a.s.
(4.2.17)
T
then f 2 L(X) and the stochastic integral is the same as the Lebesgue-Stieltjes integral. This
follows from the dominated convergence theorem and Theorem 4.21. However, L(X) may
include processes f that may not satisfy (4.2.17). We will return to this later.
Remark 4.23. Suppose X is an r.c.l.l. adapted such that
ucp
! 0.
(4.2.18)
F
f n ! 0 uniformly ) JX (f n )
A
Of course every stochastic integrator satisfies this property. Let S1 denote the class of f 2 S
(simple functions) that are bounded by 1. For T < 1 the family of random variables
Z T
{
f dX : f 2 S1 }
(4.2.19)
0
R
is bounded in probability or tight in the sense that
8" > 0 9 M < 1 such that sup P(|
f 2S1
Z
T
f dX|
0
M )  ".
(4.2.20)
D
To see this, suppose (4.2.18) is true but (4.2.20) is not true. Then we can get an " > 0 and
for each m 1, f m 2 S with |f m |  1 such that
Z T
P(|
f m dX| m) ".
(4.2.21)
0
RT
Writing g m =
it follows that g m ! 0 uniformly but in view of (4.2.21), 0 g m dX does
not converge to zero in probability- which contradicts (4.2.18). Indeed, the apparently waeker
property (4.2.18) characterises stochastic integrators as we will see later. See Theorem 5.78.
1 m
mf ,
4.3
Properties of the Stochastic Integral
First we note linearity of (f, X) 7!
R
f dX.
102
Chapter 4. Stochastic Integration
Theorem 4.24. Let X, Y be stochastic integrators, f, g be predictable processes and ↵,
R.
(i) Suppose f, g 2 L(X). Let h = ↵f + g. Then (f + g) 2 L(X) and
Z
Z
Z
hdX = ↵ f dX +
gdX.
2
(4.3.1)
(ii) Let Z = ↵X + Y. Then Z is a stochastic integrator. Further, if f 2 L(X) and
f 2 L(Y ). then f 2 L(Z) and
Z
Z
Z
f dZ = ↵ f dX +
f dY.
(4.3.2)
T
Proof. We will begin by showing that (4.3.1) is true for f, g bounded predictable processes.
For a bounded predictable process f , let
Z
Z
Z
e
K(f ) = {g 2 B(⌦, P) : (↵f + g)dX = ↵ f dX +
gdX, 8↵, 2 R}.
R
A
F
If f 2 S, it easy to see that S ✓ K(f ) and Theorem 4.20 implies that K(f ) is bp-closed.
e P).
Hence invoking Theorem 2.64, it follows that K(f ) = B(⌦,
e P) and the part proven above yields S ✓ K(f ). Once again,
Now we take f 2 B(⌦,
e P). Thus (4.3.1) is true when
using that K(f ) is bp-closed we conclude that K(f ) = B(⌦,
f, g are bounded predictable process.
Now let us fix f, g 2 L(X). We will show (|↵f | + | g|) 2 L(X), let un be bounded
predictable processes converging to u pointwise and
|un |  (|↵f | + | g|); 8n
1.
D
Let v n = un 1{|↵f || g|} and wn = un 1{|↵f |>| g|} . Then v n and wn converge pointwise to
v = u1{|↵f || g|} and w = u1{|↵f |>| g|} respectively and further
|v n |  2| g|
|wn |  2|↵f |.
Note that since v n , wn are bounded and un = v n + wn , from the part proven above, we
have
Z
Z
Z
v n dX + wn dX = un dX.
R
R
Since f, g 2 L(X), it follows that v n dX and wn dX are Cauchy in ducp metric and hence
R
so is their sum un dX. Thus (|↵f |+| g|) 2 L(X) and as a consequence, (↵f + g) 2 L(X)
as well.
4.3. Properties of the Stochastic Integral
103
Now let
f n = f 1{|f |n} , g n = g 1{|g|n} .
R
R
R
R
Then by definition, f n dX converges to f dX and g n dX converges to gdX in ducp
metric. Also (↵f n + g n ) are bounded predictable processes, converge pointwise to (↵f + g)
and is dominated by (|↵f | + | g|) 2 L(X). Hence by Theorem 4.20 we have
Z
Z
ucp
n
n
(↵f + g )dX ! (↵f + g)dX.
T
On the other hand, the validity of (4.3.1) for bounded predictable processes yields
Z
Z
Z
(↵f n + g n )dX = ↵f n dX +
g n dX
Z
Z
= ↵ f n dX +
g n dX
Z
Z
ucp
! ↵ f dX +
gdX.
F
This completes proof of (i).
For (ii), we begin by noting that (4.3.2) is true when f is simple i.e.
JZ (f ) = ↵JX (f ) + JY (f ) f 2 S.
D
R
A
e P), it follows that so does JZ and hence
Since JX , JY have a continuous extension to B(⌦,
Z is also a stochastic integrator and thus (4.3.2) is true for bounded predictable processes.
e P), converge pointwise to g, g n is
Now if f 2 L(X) and f 2 L(Y ) and g n 2 B(⌦,
R n
R n
dominated by |f |, then ↵ g dX and
g dY are Cauchy in ducp metric and hence so
R n
R
is their sum, which equals g d(↵X + Y ) = g n dZ. Thus f 2 L(Z). The equation
(4.3.2) follows by using (4.3.2) for the bounded process f n = f 1{|f |n} and passing to the
limit.
Thus the class of stochastic integrators is a linear space. We will see later that it is
R
indeed an Algebra. Let us note that when X is a continuous process, then so is f dX.
Theorem 4.25. Let X be a continuous process and further X be a stochastic integrator.
R
Then for f 2 L(X), f dX is also a continuous process.
Proof. Let
Z
e
K = {f 2 B(⌦, P) :
f dX is a continuous process}.
R
By using the definition of f dX it is easy to see that S ✓ K. Also that K is bp-closed
ucp
since Z n continuous, Z n ! Z implies Z is also continuous. Hence invoking Theorem 2.64
104
Chapter 4. Stochastic Integration
e P). The general case follows by noting that limit in ducp metric of
we conclude K = B(⌦,
R
continuous process is a continuous process and using that for f 2 L(X), f dX is the limit
R
in ducp -metric of f 1{|f |n} dX.
We can now prove :
Theorem 4.26. Dominated Convergence Theorem for the Stochastic Integral
Let X be a stochastic integrator. Suppose hn , h are predictable processes such that
h
hnt (!) !t (!) 8t
and there exists f 2 L(X) such that
Z
|hn |  |f | 8n.
n
h dX
ucp
!
Z
(4.3.3)
(4.3.4)
hdX.
T
Then
0, 8! 2 ⌦
(4.3.5)
R
A
F
Proof. Let g n = hn 1{|hn |n} and f n = hn 1{|hn |>n}
Note that in view of (4.3.4) and the assumption f 2 L(X) it follows that hn 2 L(X).
Pointwise convergence of hn to h also implies |h|  |f | which in turn yields h 2 L(X).
R
R
Thus hn dX, hdX are defined. Clearly, g n ! h pointwise and also f n ! 0 pointwise.
Further, hn = g n + f n , |g n |  |f | and |f n |  |f |.
R
ucp R
From Theorem 4.20 it follows that g n dX ! hdX and from the definition of L(X),
R
ucp
it follows that f n dX ! 0. Now linearity of the stochastic integral, Theorem 4.24, shows
that (4.3.5) is true.
D
Remark 4.27. The condition (4.3.3) that hn ! h pointwise can be replaced by requiring
that convergence holds pointwise outside a null set, namely that there exists ⌦0 ✓ ⌦ with
P(⌦0 ) = 1 such that
hnt (!) ! ht (!) 8t 0, 8! 2 ⌦0 .
(4.3.6)
See Theorem 4.13 and Corollary 4.14.
It should be noted that the hypothesis in the dominated convergence theorem given
above are exactly the same as in the case of Lebesgue integrals.
Recall, for an r.c.l.l. process X, X denotes the l.c.r.l. process defined by Xt = X(t ),
i.e. the left limit at t with the convention X(0 ) = 0 and X = X X . Note that
( X)t = 0 at each continuity point and equals the jump otherwise. Note that by the above
convention
( X)0 = X0 .
4.3. Properties of the Stochastic Integral
105
The next result connects the jumps of the stochastic integral with the jumps of the integrator.
Theorem 4.28. Let X be a stochastic integrator and let f 2 L(X). Then we have
Z
( f dX) = f · ( X).
(4.3.7)
F
T
Proof. For f 2 S, (4.3.7) can be verified from the definition. Now the class K of f such that
(4.3.7) is true can be seen to be bp-closed and hence is the class of all bounded predictable
processes. The case of general f 2 L(X) can be completed as in the proof of Theorem
4.25.
Rt
The equation (4.3.7) is to be interpreted as follows: if Yt = 0 f dX then ( Y )t =
ft ( X)t .
We have already seen that the class of stochastic integrators (with respect to a filtration
on a given probability space) is a linear space.
R
Theorem 4.29. Let X be a stochastic integrator and let f 2 L(X). Let Y = f dX. Then
Y is also a stochastic integrator and for a bounded predictable process g,
Z t
Z t
gdY =
gf dX, 8t.
(4.3.8)
0
A
0
Further, for a predictable process g, g 2 L(Y ) if and only if gf 2 L(X) and then (4.3.8)
holds.
R
Proof. Let us first assume that f, g 2 S are given by
m
X
f (s) = a0 1{0} (s) +
aj+1 1(sj ,sj+1 ] (s)
(4.3.9)
D
j=0
g(s) = b0 1{0} (s) +
n
X
bj+1 1(tj ,tj+1 ] (s)
(4.3.10)
j=0
where a0 and b0 are bounded F0 measurable random variables, 0 = s0 < s1 < s2 < . . . , <
sm+1 < 1, 0 = t0 < t1 < t2 < . . . , < tn+1 < 1; aj+1 is bounded Fsj measurable random
variable, 0  j  m and bj+1 is bounded Ftj measurable random variable, 0  j  n . Let
us put
A = {sj : 0  j  (m + 1)} [ {tj : 0  j  (n + 1)}.
Let us enumerate the set A as
A = {ri : 0  i  k}
106
Chapter 4. Stochastic Integration
where 0 = r0 < r1 < . . . < rk+1 . Note k may be smaller than m + n as there could be
repetitions. We can then represent f, g as
f (s) = c0 1{0} (s) +
k
X
cj+1 1(rj ,rj+1 ] (s)
(4.3.11)
g(s) = d0 1{0} (s) +
k
X
dj+1 1(rj ,rj+1 ] (s)
(4.3.12)
j=0
j=0
where cj+1 , dj+1 are bounded Frj measurable. Then
Z
t
(gf )dX = d0 c0 X0 +
0
k
X
j=0
Also, Y0 = c0 X0 and Yt = c0 X0 +
(Yrj+1 ^t
Pk
dj+1 cj+1 1(rj ,rj+1 ] (s)
j=0
dj+1 cj+1 (Xrj+1 ^t
j=0 cj+1 (Xrj+1 ^t
Yrj ^t ) = cj+1 (Xrj+1 ^t
A
0
R
j=0
Yt =
Z
(4.3.13)
Xrj ^t ) and hence
Xrj ^t )
Thus, using (4.3.13)-(4.3.14), one gets
Z t
k
X
(gf )dX = d0 Y0 +
dj+1 (Yrj+1 ^t
where
Xrj ^t ).
F
and hence
k
X
T
(gf )(s) = d0 c0 1{0} (s) +
(4.3.14)
Yrj ^t )
(4.3.15)
t
f dX.
(4.3.16)
0
D
R
The right hand side in (4.3.15) is gdY and thus in view of (4.3.10) we also have
Z t
n
X
(gf )dX = b0 Y0 +
bj+1 (Ytj+1 ^t Ytj ^t )
(4.3.17)
0
or in other words,
Z
0
t
(gf )dX = b0 Y0 +
j=0
n
X
j=0
bj+1 (
Z
tj+1 ^t
0
f dX
Z
tj ^t
f dX).
(4.3.18)
0
Thus the result is proven when both f, g 2 S. Now fix g 2 S. Note that this fixes n as well.
Let
e P) : (4.3.18) holds}.
K = {f 2 B(⌦,
4.3. Properties of the Stochastic Integral
107
We have seen that S ✓ K. Easy to see using Theorem 4.26 (dominated convergence
e P). Thus, for all
theorem) that K is bp- closed and since it contains S, it equals B(⌦,
bounded predictable f , (4.3.18) is true.
If f 2 L(X), then approximating f by f n = f 1{|f |n} , using (4.3.18) for f n and taking
limits, we conclude (invoking dominated convergence theorem, which is justified as g is
bounded) that (4.3.18) is true for g 2 S and f 2 L(X).
Now fix f 2 L(X) and let Y be given by (4.3.16). Note that right hand side in (4.3.18)
is JY (g)(t), as defined by (4.2.1)-(4.2.2), so that we have
Z t
(gf )dX = JY (g)(t), 8g 2 S.
(4.3.19)
0
Let us define J(g) =
Rt
0
gf dX for bounded predictable g, then J is an extension of
bp
ucp
T
JY as noted above. The Theorem 4.26 again yields that if g n ! g then J(g n ) ! J(g).
Thus, J is the extension of JY as required in the definition of stochastic integrator. Thus
Y is a stochastic integrator and (4.3.8) holds (for bounded predictable g).
F
Now suppose g is predictable such that f g 2 L(X). We will prove that g 2 L(Y ) and
that (4.3.8) holds for such a g.
A
To prove g 2 L(Y ), let hk be bounded predictable, converging pointwise to h such that
R
hk are dominated by g. We need to show that hk dY is Cauchy in ducp metric.
D
R
Let uk = hk f . Then uk are dominated by f g 2 L(X) and converge pointwise to hf .
R
R
Thus, Z k = uk dX converges to Z = hf dX and hence is Cauchy in ducp metric. On the
R
other hand, since hk is bounded, invoking (4.3.8) for hk , we conclude hk dY = Z k and
hence is Cauchy in ducp metric. This shows g 2 L(Y ). Further, taking h = g, hk = g 1{|g|k}
R
R
R
above we conclude that the limit of hk dY is gdY which in turn equals hf dX as noted
above. This also shows that (4.3.8) is true when f g 2 L(X) and g 2 L(Y ).
To complete the proof, we need to show that if g 2 L(Y ) then f g 2 L(X). For this,
suppose un are bounded predictable, |un |  |f g| and un converges to 0 pointwise. Need to
R
ucp
show un dX ! 0. Let
8
< 1
if fs (!) 6= 0
˜
fs (!) = fs (!)
:0
if f (!) = 0.
s
and v n = un f˜. Now v n are predictable, are dominated by |g| and v n converges pointwise
to 0. Thus,
Z
ucp
v n dY ! 0.
108
Chapter 4. Stochastic Integration
Since |un |  |f g|, it follows that v n f = un and thus v n f is bounded and hence in L(X).
Thus by the part proven above, invoking (4.3.8) for v n we have
Z
Z
Z
n
n
v dY = v f dX = un dX.
This shows
R
un dX
ucp
! 0. Hence f g 2 L(X). This completes the proof.
We had seen in Theorem 4.3 that the class of predictable processes is essentially the
same for the filtrations F⇧ and F⇧+ - the only di↵erence being at t = 0.
We now observe :
T
Theorem 4.30. For an r.c.l.l. F⇧ adapted process X, it is a stochastic integrator w.r.t. the
filtration (F⇧ ) if and only if it is a stochastic integrator w.r.t. the filtration (F⇧+ ).
F
R
Proof. Let X be a stochastic integrator w.r.t. the filtration (F⇧ ), so that hdX is defined
for bounded (F⇧ ) predictable processes h. Given a bounded (F⇧+ )-predictable processes f ,
let g be defined by
A
gt (!) = ft (!)1{(0,1)⇥⌦} (t, !).
R
Then by Theorem 4.3, g is a bounded (F⇧ ) predictable processes. So we define
Z
JX (f ) = gdX + f0 X0 .
D
It is easy to check that JX satisfies the required properties for X to be a Stochastic
integrator.
R
Conversely if X is a stochastic integrator w.r.t. the filtration (F⇧+ ), so that hdX is
defined for bounded (F⇧+ )-predictable processes h, of course for a bounded (F⇧ ) predictable
R
f we can define by JX (f ) = f dX and JX will have the required continuity properties.
However, we need to check that JX (f ) so defined is (F⇧ ) adapted or in other words, belongs
to R0 (⌦, (F⇧ ), P). For f 2 S, it is clear that JX (f ) 2 R0 (⌦, (F⇧ ), P) since X is (F⇧ ) adapted.
Since the space R0 (⌦, (F⇧ ), P) is a closed subspace of R0 (⌦, (F⇧+ ), P) in the ducp metric,
e P(F⇧ )) and thus X is a stochastic
it follows that JX (f ) 2 R0 (⌦, (F⇧ ), P) for f 2 B(⌦,
integrator for the filtration (F⇧ ).
Exercise 4.31. Let X be a (F⇧ )- stochastic integrator. For each t let {Gt : t
0} be a
+
filtration such that for all t, Ft ✓ Gt ✓ Ft . Show that X is a (G⇧ )- stochastic integrator.
4.4. Locally Bounded Processes
4.4
109
Locally Bounded Processes
We will introduce an important class of integrands, namely that of locally bounded predictable processes, that is contained in L(X) for every stochastic integrator X. For a stop
e : 0  t  ⌧ (!)} and thus g = f 1[0,⌧ ] means
time ⌧ , [0, ⌧ ] will denote the set {(t, !) 2 ⌦
the following: gt (!) = ft (!) if t  ⌧ (!) and gt (!) is zero if t > ⌧ (!).
The next result gives interplay between stop times and stochastic integration.
T
Lemma 4.32. Let X be a stochastic integrator and f 2 L(X). Let ⌧ be a stop time. Let
g = f 1[0,⌧ ] . Let
Z t
Yt =
f dX
(4.4.1)
0
R
and V = gdX. Then Vt = Yt^⌧ , i.e.
Z t
Yt^⌧ =
f 1[0,⌧ ] dX.
(4.4.2)
0
R
A
F
Proof. When f 2 S is a simple predictable process and ⌧ is a stop time taking only finitely
many values, then g 2 S and (4.4.1) - (4.4.2) can be checked as in that case, the integrals
R
R
f dX and gdX are both defined directly by (4.2.2). Thus fix f 2 S. Approximating a
bounded stop time ⌧ from above by stop time taking finitely many values (as seen in the
proof of Theorem 2.52), it follows that (4.4.2) is true for any bounded stop time, then any
stop time ⌧ can be approximated by ⌧˜n = ⌧ ^ n and one can check that (4.4.2) continues
to be true. Thus we have proven the result for simple integrands.
Now fix a stop time ⌧ and let
e P) : (4.4.1)
K = {f 2 B(⌦,
(4.4.2) is true for all t
0.}.
D
Then it is easy to see that K is closed under bp-convergence and as noted above it contains
e P). Finally, for a general f 2 L(X),
S. Hence by Theorem 2.64, it follows that K = B(⌦,
the result follows by approximating f by f n = f 1{|f |n} and using dominated convergence
theorem. This completes the proof.
Exercise 4.33. In the proof given above, we first proved the required result for f 2 S and
any stop time ⌧ . Complete the proof by first fixing a simple stop time ⌧ and prove it for all
f 2 L(X) and subsequently prove it for all stop times ⌧ .
R t^⌧
Remark 4.34. We can denote Yt^⌧ as 0 f dX so that (4.4.2) can be recast as
Z t^⌧
Z t
f dX =
f 1[0,⌧ ] dX.
(4.4.3)
0
0
110
Chapter 4. Stochastic Integration
Corollary 4.35. If X is a stochastic integrator and f, g 2 L(X) and ⌧ is a stop time
such that
f 1[0,⌧ ] = g 1[0,⌧ ]
then for each t
Z
t^⌧
f dX =
0
Z
t^⌧
gdX.
(4.4.4)
0
Definition 4.36. A process f is said to be locally bounded if there exist stop times ⌧ n ,
0  ⌧ 1  ⌧ 2  . . ., ⌧ n " 1 such that for every n,
f 1[0,⌧ n ] is bounded.
The sequence {⌧ n : n
1} is called a localizing sequence.
sup
0t⌧n (!)
|ft (!)|  Cn ) = 1, 8n
1.
(4.4.5)
F
P(! :
T
Thus if a process f is locally bounded, then we can get stop times ⌧ n increasing to
infinity (can even choose each ⌧ n to be bounded) and constants Cn such that
A
Note that given finitely many locally bounded processes one can choose a common
localizing sequence {⌧ n : n
1}. A continuous adapted process X such that X0 is
bounded is easily seen to be locally bounded. We can take the localizing sequence to be
⌧n = inf{t
0 : |X(t)|
n or t
n}.
R
For an r.c.l.l. adapted process X, recall that X is the process defined by X (t) =
X(t ), where X(t ) is the left limit of X(s) at s = t for t > 0 and X (0) = X(0 ) = 0.
Let ⌧n be the stop times defined by
D
⌧n = inf{t
0 : |X(t)|
n or |X(t )|
n or t
n}.
(4.4.6)
Then it can be easily seen that X 1[0,⌧n ] is bounded by n and that ⌧n " 1 and hence
X is locally bounded. Easy to see that sum of locally bounded processes is itself locally
bounded. Further, if X is a r.c.l.l. process with bounded jumps : | X|  K, then X is
locally bounded, since X is locally bounded and ( X) is bounded and X = X + ( X).
Exercise 4.37. Let X be an r.c.l.l. adapted process. Show that X is locally bounded if and
only if X is locally bounded.
For future reference, we record these observations as a Lemma.
Lemma 4.38. Let X be an adapted r.c.l.l. process.
4.4. Locally Bounded Processes
111
(i) If X is continuous with X0 bounded, then X is locally bounded.
(ii) X
(iii) If
is locally bounded.
X is bounded, then X is locally bounded.
We now prove an important property of the class L(X).
Theorem 4.39. Let X be an integrator and f be a predictable process such that there exist
stop times ⌧m increasing to 1 with
f 1[0,⌧m ] 2 L(X) 8n
1.
(4.4.7)
Then f 2 L(X).
T
Proof. Let g n be bounded predictable such that g n ! g pointwise and |g n |  |f |. Let
R
Z n = g n dX. Now for each m,
(4.4.8)
|g n 1[0,⌧m ] |  |f 1[0,⌧m ] |
(4.4.9)
and Y n,m defined by
=
Z
g n 1[0,⌧m ] dX
A
Y
n,m
F
g n 1[0,⌧m ] ! g 1[0,⌧m ] pointwise,
satisfies
(4.4.11)
R
n
Ytn,m = Zt^⌧
.
m
(4.4.10)
D
In view of (4.4.8), (4.4.9) the assumption (4.4.7) implies that for each m, {Y n,m : n 1}
is Cauchy in ducp metric. Thus invoking Corollary 2.73, we conclude that Z n is Cauchy in
ducp metric and hence f 2 L(X).
As noted earlier, f 2 L(X), h predictable, |h|  C|f | for some constant C > 0 implies
h 2 L(X). Thus the previous result gives us
Corollary 4.40. Let X be a stochastic integrator, g be a locally bounded predictable
process and f 2 L(X). Then f g belongs to L(X).
In particular, we have
Corollary 4.41. Let g be a locally bounded predictable process. Then g belongs to L(X)
for every stochastic integrator X. And if Y is an r.c.l.l. adapted process, then Y 2 L(X)
for every stochastic integrator X.
112
Chapter 4. Stochastic Integration
Exercise 4.42. Let X be a stochastic integrator. Let s0 = 0 < s1 < s2 < . . . < sn < . . .
with sn " 1. Let ⇠j , j = 1, 2 . . . , be such that ⇠j is Fsj 1 measurable.
P
(i) For n 1 let hn = nj=1 ⇠j 1(sj 1 ,sj ] . Show that hn 2 L(X) and
Z t
n
X
n
h dX =
⇠j (Xsj ^t Xsj 1 ^t ).
0
j=1 ⇠j 1(sj
1 ,sj ]
j=1
. Show that h 2 L(X) and
Z t
1
X
hdX =
⇠j (Xsj ^t Xsj
0
j=1
For an r.c.l.l. process X and a stop time , let X [
defined as follows
[ ]
Xt = Xt^ .
]
1 ^t
).
denote the process X stopped at
T
(ii) Let h =
P1
F
Next result shows that if X is a stochastic integrator and
is a stochastic integrator as well.
is a stop time, then Y = X [
A
Lemma 4.43. Let X be a stochastic integrator and be a stop time. Then X [
Rt
stochastic integrator and for f 2 L(X), writing Zt = 0 f dX, one has
Z t
Z t
[ ]
Zt^ =
f dX =
(f 1[0, ] )dX.
0
(4.4.12)
]
]
is also a
(4.4.13)
0
R
Proof. First one checks that (4.4.13) is true for f 2 S. Then for any bounded predictable
f , defining the process
Z
J0 (f ) = (f 1[0, ] )dX
bp
ucp
D
one can check that fn ! f implies J0 (fn ) ! J0 (f ) and hence it follows that X [ ]
is a stochastic integrator. Using (4.4.4), it follows that (4.4.13) is true for all bounded
predictable processes f . Finally, for a general f 2 L(X), the result follows by approximating f by f n = f 1{|f |n} and using dominated convergence theorem. This completes the
proof.
R
Exercise 4.44. Deduce the previous result using Theorem 4.29 by identifying X [ ] as gdX
for a suitable g 2 L(X).
The next result shows that if we localize the concept of integrator, we do not get
anything new- i.e. if a process is locally a stochastic integrator, then it is already a stochastic
integrator.
4.4. Locally Bounded Processes
113
Theorem 4.45. Suppose X is an adapted r.c.l.l. process such that there exist stop times
n
⌧ n such that ⌧ n  ⌧ n+1 for all n and ⌧ n " 1 and the stopped processes X n = X [⌧ ] are
stochastic integrators. Then X is itself a stochastic integrator.
R
e P) and for m 1 let U m = f dX m . Without loss of generality we
Proof. Fix f 2 B(⌦,
k
assume that ⌧ 0 = 0. Then using (4.4.4), it follows that Utm = Ut^⌧
m for m  k. We define
J0 (f ) by J0 (f )0 = f0 X0 and for m 1,
⌧m
It follows that
J0 (f )t^⌧ m =
Of course, for simple predictable f , J0 (f ) =
Z
1
t
< t  ⌧ m.
f dX m .
(4.4.14)
(4.4.15)
0
Rt
0
f dX and thus J0 is an extension of JX .
bp
T
J0 (f )t = Utm ,
1 we
F
Now let f n be bounded predictable such that f n ! f . Using (4.4.15), for n, m
have
Z t
J0 (f n )t^⌧ m =
f n dX m .
0
(f n )
Writing
= J0
and Z = J0 (f ), it follows using (4.4.15) that Z n , n
1 and Z
ucp
n
satisfy(2.5.9) and hence by Lemma 2.72 it follows that Z
! Z. We have thus proved
bp
ucp
n
n
that f ! f implies J0 (f ) ! J0 (f ) and since J0 (f ) agrees with JX (f ) for simple
predictable f , it follows that X is a stochastic integrator.
A
Zn
R
We have seen a version of the dominated convergence theorem for stochastic integration.
Here is another result on convergence that will be needed later.
D
Theorem 4.46. Suppose Y n , Y 2 R0 (⌦, (F⇧ ), P), Y n
grator. Then
Z
Z
(Y n ) dX
ucp
!
ucp
! Y and X is a stochastic inte-
Y dX.
and (Y n ) belong to L(X). Let
Z
Z
n
bn = ducp ( (Y ) dX, Y dX).
Proof. We have noted that Y
To prove that bn ! 0 suffices to prove the following: For any subsequence {nk : k
1},
there exists a further subsequence {mj : j
1} of {nk : k
1} (i.e. 9 subsequence
{kj : j 1} such that mj = nkj ) such that
bmj ! 0.
(4.4.16)
114
Chapter 4. Stochastic Integration
So now, given a subsequence {nk : k 1}, using ducp (Y nk , Y ) ! 0, let us choose mj = nkj
with kj+1 > kj and ducp (Y mj , Y )  2 j . Then as seen in the proof of Theorem 2.68, this
would imply
1
X
m
[sup|Yt j Yt |] < 1, 8T < 1.
j=1 tT
Thus defining
Ht =
1
X
j=1
mj
| Yt
Yt |
(4.4.17)
T
it follows that (outside a fixed null set ) the convergence in (4.4.17) is uniform on t 2 [0, T ]
for all T < 1 and as a result H is an r.c.l.l. adapted process. Thus the processes (Y mj )
are dominated by (H + Y ) which is a locally bounded process, as H + Y is an r.c.l.l.
adapted process. Thus the dominated convergence Theorem 4.26 yields
Z
Z
bmj = ducp ( (Y mj ) dX, Y dX) ! 0.
ucp
! Y , then there is a subsequence that is dominated by a
A
Exercise 4.47. Show that if Y n
locally bounded process.
F
This completes the proof as explained above.
R
The subsequence technique used in the proof also yields the following result, which will
be useful alter.
ucp
D
Proposition 4.48. Let Y n ! Y , where Y n , Y are Rd -valued r.c.l.l. processes and g n , g :
[0, 1) ⇥ Rd 7! R be continuous functions such that g n converges to g uniformly on compact
ucp
subsets of [0, 1) ⇥ Rd . Let Ztn = g n (t, Ytn ) and Zt = g(t, Yt ). Then Z n ! Z.
Proof. Like in the previous proof, let bn = ducp (Z n , Z) and given any subsequence {nk :
k 1}, using ducp (Y nk , Y ) ! 0, choose mj = nkj with kj+1 > kj and ducp (Y mj , Y )  2 j .
It follows that
1
X
m
[ sup|Yt j Yt | ] < 1, 8T < 1.
j=1 tT
m
and so [ suptT |Yt j Yt |] converges to zero for all T < 1 a.s. and now uniform convergence
of g n to g on compact subsets would yield convergence of Z mj to Z uniformly on [0, T ] for
all T and thus bmj = ducp (Z mj , Z) converges to 0. Thus every subsequence of {bn } has a
further subsequence converging to zero and hence limn!1 bn = 0.
4.5. Approximation by Riemann Sums
115
Essentially the same proof as given above for Theorem 4.46 gives the following result,
only di↵erence being that if X 2 V i.e. if Var[0,t] (X) < 1 for all t < 1, and Y is an r.c.l.l.
R
R
adapted process, the integral Y dX is defined in addition to Y dX, both are defined
as Lebesgue-Stieltjes integrals while the later agrees with the stochastic integral (as seen
in Theorem 4.21).
ucp
Proposition 4.49. Suppose Y n , Y 2 R0 (⌦, (F⇧ ), P), Y n
with finite variation paths). Then
Z
Z
ucp
n
(Y )dX ! Y dX.
Approximation by Riemann Sums
T
4.5
! Y and X 2 V (a process
F
The next result shows that for an r.c.l.l. process Y and a stochastic integrator X, the
R
stochastic integral Y dX can be approximated by Riemann like sums, with one di↵erencethe integrand must be evaluated at the lower end point of the interval as opposed to any
point in the interval in the Riemann-Stieltjes integral.
Theorem 4.50. Let Y be an r.c.l.l. adapted process and X be a stochastic integrator. Let
tm
n " 1 as n " 1
A
m
m
0 = tm
0 < t1 < . . . < tn < . . . ;
(4.5.1)
be a sequence of partitions of [0, 1) such that for all T < 1,
=(
sup
{n : tm
n T }
R
m (T )
Let
Ztm =
1
X
n=0
(tm
n+1
tm
n )) ! 0 as m " 1.
Ytm
(Xtm
n ^t
n+1 ^t
Xtm
)
n ^t
(4.5.2)
(4.5.3)
D
R
and Z = Y dX. Note that for each t, m, the sum in (4.5.3) is a finite sum since
tm
n ^ t = t from some n onwards. Then
Zm
or in other words
1
X
n=0
Ytm
(X
n ^t
tm
n+1 ^t
ucp
!Z
Xtm
)
n ^t
(4.5.4)
ucp
!
Z
t
Y dX.
0
Proof. Let Y m be defined by
Ytm
=
1
X
n=0
Ytm
1 m m (t).
n ^t (tn , tn+1 ]
116
Chapter 4. Stochastic Integration
We will first prove
Z
Y m dX = Z m .
(4.5.5)
For this, let Vt = supst |Y |. Then V is an r.c.l.l. adapted process and hence V
bounded.
k (x)
Let us fix m and let
= max(min(x, k), k), so that |
Utk
=
k
X
n=0
k
k (x)|
is locally
 k for all x 2 R. Let
m
(Ytm
)1(tm
(t).
n ^t
n ,tn+1 ]
0
converges to
m
R
Y
m dX.
T
Then U k 2 S and U k converges pointwise (as k increases to 1) to Y m . Further, |Utk | 
Vt = V (t ) for all t and hence by the Dominated Convergence Theorem 4.26
Z t
k
X
k
U k dX =
(Ytm
)(Xtm
Xtm
)
n ^t
n ^t
n+1 ^t
n=0
This proves (4.5.5).
which is same as (4.5.4).
A
F
Now, Y converges pointwise to Y and are dominated by the locally bounded process
V . Hence again by Theorem 4.26,
Z
Z
ucp
m
Y dX ! Y dX
R
We will next show that the preceding result is true when the sequence of deterministic
partitions is replaced by a sequence of random partitions via stop times. For this, we need
the following lemma.
D
Lemma 4.51. Let X be a stochastic integrator, Z be an r.c.l.l. adapted process and ⌧ be
stop time. Let
h = Z⌧ 1(⌧,1) .
Then h is locally bounded predictable and
Z t
hdX = Z⌧ ^t (Xt
0
(4.5.6)
X⌧ ^t ).
Proof. We have seen in Proposition 4.1 that h is predictable. Since
sup |hs |  sup |Zs |
0st
and Z
0st
is locally bounded, it follows that h is locally bounded and thus h 2 L(X).
(4.5.7)
4.5. Approximation by Riemann Sums
117
If Z is a bounded r.c.l.l. adapted process and ⌧ takes finitely many values, then easy to
see that h belongs to S and that (4.5.7) is true. Now if ⌧ is a bounded stop time, then for
m 1, ⌧ m defined by
⌧ m = 2 m ([2m ⌧ ] + 1)
(4.5.8)
are stop times, each taking finitely many values and ⌧ m # ⌧ . One can then verify (4.5.7) by
approximating ⌧ by ⌧ m defined via (4.5.8) and then using the fact that hm = Z⌧ m 1(⌧ m ,1)
converges boundedly pointwise to h, validity of (4.5.7) for ⌧ m implies the same for ⌧ . For
a general ⌧ , we approximate it by ⌧n = ⌧ ^ n. Thus, it follows that (4.5.6)-(4.5.7) are true
for bounded r.c.l.l. adapted processes Z. For a general Z, let
Z n = max(min(Z, n), n).
T
Noting that hn = Z⌧n 1(⌧,1) converges to h and |hn |  |h|, the validity of (4.5.7) for Z n
implies the same for Z.
be another stop time
F
Corollary 4.52. Let Z and ⌧ be as in the previous lemma and
with ⌧  . Let
g = Z⌧ 1(⌧, ] .
Z
t
0
A
Then
gdX = Z⌧ ^t (Xt^
Xt^⌧ ) = Z⌧ (Xt^
Xt^⌧ ).
(4.5.9)
(4.5.10)
R
The first equality follows from the observation that g = h1[0, ] where h is as in Lemma
Rt
R·
4.51 and hence 0 gdX = ( 0 hdX)t^ (using Lemma 4.32). The second equality can be
directly verified.
]
as 1(⌧,1)
1(
,1)
and thereby deduce the above Corollary from
D
Exercise 4.53. Express 1(⌧,
Lemma 4.51.
Corollary 4.54. Let X be a stochastic integrator and ⌧ 
F⌧ measurable random variable. Let
be stop times. Let ⇠ be a
f = ⇠ 1(⌧, ] .
Then f 2 L(X) and
Z
t
f dX = ⇠(Xt^
Xt^⌧ ).
0
Proof. This follows from the Corollary 4.52 by taking Z = ⇠ 1[⌧,1) and noting that as shown
in Lemma 2.39, Z is adapted.
118
Chapter 4. Stochastic Integration
Definition 4.55. For > 0, a -partition for an r.c.l.l. adapted process Z is a sequence of
stop times {⌧n ; : n 0} such that 0 = ⌧0 < ⌧1 < . . . < ⌧n < . . . ; ⌧n " 1 and
|Zt
Z ⌧n | 
for ⌧n  t < ⌧n+1 , n
0.
(4.5.11)
Remark 4.56. Given r.c.l.l. adapted processes Z i , 1  i  k and > 0, we can get a
sequence of partitions {⌧n : n
0} such that {⌧n : n
0} is a partition for each of
Z 1 , Z 2 , . . . Z k . Indeed, let {⌧n : n 0} be defined inductively via ⌧0 = 0 and
⌧n+1 = inf{t > ⌧n : max( max |Zti
1ik
Z⌧in |, max |Zti
1ik
Invoking Theorem 2.44, we can see that {⌧n : n
Z⌧in |)
}.
(4.5.12)
0} are stop times and that limn"1 ⌧n = 1.
n=0
Zt | 
m
and hence Z m
k
X
Ztm,k =
ucp
! Z . For k
1, let
F
Then it follows that
|Ztm
T
Let m # 0 and for each m, let {⌧nm : n 0} be a m -partition for Z. We implicitly
assume that m > 0. Let
1
X
m
m ] (t).
Zt =
Zt^⌧nm 1(⌧nm ,⌧n+1
m ] (t).
Zt^⌧nm 1(⌧nm ,⌧n+1
n=0
A
Now, using Corollary 4.52 and linearity of stochastic integrals, we have
Z t
k
X
m ^t
Z m,k dX =
Zt^⌧nm (X⌧n+1
X⌧nm ^t ).
0
n=0
D
R
Let Vt = supst |Zs |. Then V is r.c.l.l. and thus as noted earlier, V is locally bounded.
Easy to see that |Zm,k |  V and Z m,k converges pointwise to Z m . Hence by Theorem
Rt
ucp R t
4.26, 0 Z m,k dX ! 0 Z m dX. Thus
Z t
1
X
m
m ^t
Z dX =
Zt^⌧nm (X⌧n+1
X⌧nm ^t ).
(4.5.13)
0
Zm
Since
converges pointwise to Z
Theorem 4.26 it follows that
Z
n=0
and |Z n |  V
m
Z dX
ucp
!
Z
with V
Z dX.
locally bounded, invoking
(4.5.14)
We have thus proved another version of Theorem 4.50.
Theorem 4.57. Let X be a stochastic integrator. Let Z be an r.c.l.l. adapted process. Let
1 let {⌧nm : n 1} be a m -partition for Z. Then
m # 0 and for m
Z t
1
X
ucp
m
Zt^⌧nm (X⌧n+1 ^t X⌧nm ^t ) !
Z dX.
(4.5.15)
n=0
0
4.6. Quadratic Variation of Stochastic Integrators
119
P
Remark 4.58. When m ( m )2 < 1, say m = 2 m , then the convergence in (4.5.15) is
stronger: it is uniform convergence on [0, T ] almost surely for each T < 1. We will prove this
in Chapter (6).
4.6
Quadratic Variation of Stochastic Integrators
We now show that stochastic integrators also, like Brownian motion, have finite quadratic
variation.
Theorem 4.59. Let X be a stochastic integrator. Then there exists an adapted increasing
process A, written as [X, X], such that
Z t
2
2
Xt = X0 + 2
X dX + [X, X]t , 8t.
(4.6.1)
1 let {⌧nm : n
# 0 and for m
1
X
n=0
Proof. For a, b 2 R,
m -partition
ucp
X⌧nm ^t )2
m ^t
(X⌧n+1
b2
1} be a
! [X, X]t .
F
m
a2 = 2a(b
for X. Then one
(4.6.2)
a)2 .
a) + (b
A
Further, Let
has
T
0
m ^t and a = X⌧ m ^t and summing with respect to n, we get
Using this with b = X⌧n+1
n
R
Xt2 = X02 + 2Vtm + Qm
t
where
Vtm =
1
X
D
n=0
and
Qm
t =
m ^t
X⌧nm ^t (X⌧n+1
1
X
n=0
X⌧nm ^t )
X⌧nm ^t )2 .
m ^t
(X⌧n+1
m ^t = Xt . In
Note that after some n that may depend upon ! 2 ⌦, ⌧nm > t and hence X⌧n+1
view of this, the two sums above have only finitely many non-zero terms.
ucp R
By Theorem 4.57, V m ! X dX. Hence, writing
Z t
2
2
At = X t X 0 2
X dX,
(4.6.3)
0
we conclude
Qm
t
ucp
! At .
120
Chapter 4. Stochastic Integration
This proves (4.6.1). Remains to show that At is an increasing process. Fix ! 2 ⌦, s  t,
m , then |X
and note that if ⌧jm  s < ⌧j+1
X⌧jm |  m and
s
Qm
s
=
j 1
X
m
(X⌧n+1
X⌧nm )2 + (Xs
X⌧jm )2
n=0
and
Qm
t =
j 1
X
X⌧nm )2 +
m
(X⌧n+1
n=0
Thus
Since Qm
1
X
n=j
m
Qm
s  Qt +
ucp
m ^t
(X⌧n+1
X⌧nm ^t )2 .
2
m.
(4.6.4)
! A, it follows that A is an increasing process.
T
Remark 4.60. From the identity (4.6.1), it follows that the process [X, X] does not depend
upon the choice of partitions {⌧nm : n 1}.
F
Definition 4.61. For a stochastic integrator X, the process [X, X] obtained in the previous
theorem is called the Quadratic Variation of X.
A
From the definition of quadratic variation, it follows that for a stochastic integrator X
and a stop time
[X [ ] , X [ ] ]t = [X, X]t^ 8t.
(4.6.5)
D
R
For stochastic integrators X, Y , let us define cross-quadratic variation between X, Y
via the polarization identity
1
[X, Y ]t = ([X + Y, X + Y ]t [X Y, X Y ]t )
(4.6.6)
4
By definition [X, Y ] 2 V since it is defined as di↵erence of two increasing processes. Also,
it is easy to see that [X, Y ] = [Y, X].
By applying Theorem 4.59 to X + Y and X Y and using that the mapping (f, X) 7!
R
f dX is bilinear one can deduce the following result.
Theorem 4.62. (Integration by Parts Formula) Let X, Y be stochastic integrators.
Then
Z t
Z t
Xt Yt = X0 Y0 +
Y dX +
X dY + [X, Y ]t , 8t.
(4.6.7)
0
Let
m
0
# 0 and for m 1 let {⌧nm : n 1} be a m -partition for X and Y . Then one has
1
X
ucp
m ^t
m ^t
(X⌧n+1
X⌧nm ^t )(Y⌧n+1
Y⌧nm ^t ) ! [X, Y ]t .
(4.6.8)
n=0
4.6. Quadratic Variation of Stochastic Integrators
121
Remark 4.63. Like (4.6.5), one also has for any stop time
and stochastic integrators X, Y :
[X [ ] , Y ]t = [X, Y [ ] ]t = [X [ ] , Y [ ] ]t = [X, Y ]t^
8t.
(4.6.9)
This follows easily from (4.6.8).
Exercise 4.64. For stochastic integrators X, Y and stops
[X [ ] , Y [⌧ ] ] = [X, Y ][
^⌧ ]
and ⌧ show that
.
Corollary 4.65. Let X, Y be stochastic integrators. Then
(ii)
(iii)
[X, X]t = (( X)t )2 , for t > 0.
P
0<st ((
X)s )2  [X, X]t < 1.
[X, Y ]t = ( X)t ( Y )t for t > 0.
T
(i)
(iv) If X (or Y ) is a continuous process, then [X, Y ] is also a continuous process.
Using b2
a2
2a(b
Xt2 = 2Xt (Xt
a) = (b
(Xt
Xt ) + [X, X]t
[X, X]t .
a)2 , we get
A
Xt2
F
Proof. For (i), using (4.6.1) and (4.3.7), we get for every t > 0
Xt )2 = [X, X]t
[X, X]t
R
which is same as (i). (ii) follows from (i) as [X, X]t is an increasing process. (iii) follows
from (i) via the polarization identity (4.6.6). And lastly, (iv) is an easy consequence of
(iii).
D
Remark 4.66. Let us note that by definition, [X, X] 2 V+
0 and [X, Y ] 2 V0 i.e. [X, X]0 = 0
and [X, Y ]0 = 0.
The next result shows that for a continuous process A 2 V, [X, A] = 0 for all stochastic
integrators X.
Theorem 4.67. Let X be a stochastic integrator and A 2 V, i.e. an r.c.l.l. process with
finite variation paths. Then
X
[X, A]t =
( X)s ( A)s .
(4.6.10)
0<st
In particular, if X, A have no common jumps, then [X, A] = 0. This is clearly the case if
one of the two processes is continuous.
122
Chapter 4. Stochastic Integration
Proof. Since A 2 V, for {⌧nm } as in Theorem 4.62 above one has
Z t
1
X
ucp
m ^t
X⌧nm ^t (A⌧n+1
A⌧nm ^t ) !
X dA
0
n=0
and
1
X
X
n=0
m ^t
⌧n+1
(A
m ^t
⌧n+1
Using (4.6.8) we conclude
[X, A]t =
and hence (4.6.10) follows.
Z
A
⌧nm ^t
t
X dA
0+
)
Z
ucp
!
Z
t
X dA.
0+
t
X dA
0
j(X, Y )t =
X
T
For stochastic integrators X, Y , let
( X)s ( Y )s .
0<st
F
The sum above is absolutely convergent in view of part (ii) Corollary 4.65. Clearly,
j(X, X) is an increasing process. Also we have seen that
j(X, X)t  [X, X]t .
A
(4.6.11)
R
We can directly verify that j(X, Y ) satisfies the polarization identity
1
[j(X, Y )t = (j(X + Y, X + Y )t j(X Y, X Y )t ).
(4.6.12)
4
The identity (4.6.7) characterizes [X, Y ]t and it shows that (X, Y ) 7! [X, Y ]t is a
bilinear form. The relation (4.6.8) also yields the parallelogram identity for [X, Y ]:
D
Lemma 4.68. Let X, Y be stochastic integrators. Then we have
[X + Y, X + Y ]t + [X
Y, X
Y ]t = 2([X, X]t + [Y, Y ]t ), 8t
0.
(4.6.13)
Proof. Let m # 0 and for m
1 let {⌧nm : n
1} be a m -partition for X. and take
m
m
m ^t
m ^t
an = (X⌧n+1
X⌧nm ^t ), bn = (Y⌧n+1
Y⌧nm ^t ). Use the identity (a + b)2 + (a b)2 =
m
2(a2 + b2 ) with a = am
n and b = bn ; sum over n and take limit over m. We will get the
required identity by (4.6.2).
p
Here is an analogue of the inequality |a2 b2 |  2(a b)2 (a2 + b2 ).
Lemma 4.69. Let X, Y be stochastic integrators. Then we have
p
|[X, X]t [Y, Y ]t |  2[X Y, X Y ]t ([X, X]t + [Y, Y ]t ).
(4.6.14)
4.6. Quadratic Variation of Stochastic Integrators
123
m
Proof. Let ⌧nm , am
n , bn be as in the proof of Lemma 4.68 above. Then we have
X
X
X
2
2
2
2
| (am
(bm
|(am
(bm
n)
n) |
n)
n) |
n
n
n
Xp

2(am
n
2
m 2
m 2
bm
n ) ((an ) + (bn ) )
n

s
2
X
(am
n
2
bm
n)
n
s
X
2
m 2
((am
n ) + (bn ) )
n
and taking limit over m, using (4.6.2), we get the required result (4.6.14).
Also, using that
[aX + bY, aX + bY ]t
|[X, Y ]t | 
p
[X, X]t [Y, Y ]t .
T
we can deduce that
0 8a, b 2 R
F
Indeed, one has to do this carefully (in view of the null sets lurking around). We can prove
a little bit more.
A
Theorem 4.70. Let X, Y be stochastic integrators. Then for any s  t
p
Var(s,t] ([X, Y ])  ([X, X]t [X, X]s ).([Y, Y ]t [Y, Y ]s ),
p
Var[s,t] ([X, Y ])  ([X, X]t [X, X]s ).([Y, Y ]t [Y, Y ]s ),
and
p
[X, X]t [Y, Y ]t ,
p
Var[0,t] (j(X, Y ))  [X, X]t [Y, Y ]t .
Var[0,t] ([X, Y ]) 
and
D
⌦a,b,s,r = {! 2 ⌦ : [aX + bY, aX + bY ]r (!)
(4.6.16)
(4.6.17)
R
Proof. Let
(4.6.15)
(4.6.18)
[aX + bY, aX + bY ]s (!)}
⌦0 = [{⌦a,b,s,r : s, r, a, b 2 Q, r
s}.
Then it follows that P(⌦0 ) = 1 (since for any process Z, [Z, Z] is an increasing process).
For ! 2 ⌦0 , for 0  s  r, s, r, a, b 2 Q
(a2 ([X, X]r
[X, X]s ) + b2 ([Y, Y ]r
[Y, Y ]s ) + 2ab([X, Y ]r
[X, Y ]s ))(!)
0.
Since the quadratic form above remains positive, we conclude
|([X, Y ]r (!)
[X, Y ]s (!))|
p
 ([X, X]r (!) [X, X]s (!))([Y, Y ]r (!)
[Y, Y ]s (!)).
(4.6.19)
124
Chapter 4. Stochastic Integration
Since all the process occurring in (4.6.19) are r.c.l.l., it follows that (4.6.19) is true for
all s  r, s, r 2 [0, 1). Taking s = 0, this proves (4.6.21). Now given s < t and
s = t0 < t1 < . . . < tm = t, we have
m
X1
|[X, Y ]tj+1 [X, Y ]tj |
j=0

m
X1 q
([X, X]tj+1
[X, X]tj )([Y, Y ]tj+1
[Y, Y ]tj )
(4.6.20)
j=0
p
 ([X, X]t
[X, X]s )([Y, Y ]t
[Y, Y ]s )
T
where the last step follows from Cauchy-Schwarz inequality using the fact that [X, X], [Y, Y ]
are increasing processes. Now taking supremum over partitions of [s, t] in (4.6.20) we get
(4.6.15). For (4.6.16), recalling definition of Var[a,b] (G) we have
Var[s,t] ([X, Y ]) = Var(s,t] ([X, Y ]) + |( [X, Y ])s |
[X, X]s )([Y, Y ]t
[Y, Y ]s ) + |( X)s ( Y )s |
[X, X]s + ( X)2s )([Y, Y ]t
F
p
([X, X]t
p
 ([X, X]t
p
 ([X, X]t

[X, X]s )([Y, Y ]t
[Y, Y ]s + ( Y )2s )
[Y, Y ]s ).
A
Now (4.6.17) follows from (4.6.16) taking s = 0. As for (4.6.18), note that
X
Var[0,t] (j(X, Y )) =
|( X)s ( Y )s |
0<st

p
0<st
( X)2s
X
( Y )2s
0<st
[X, X]t [Y, Y ]t .
D
R

sX
Corollary 4.71. For stochastic integrators X, Y , one has
p
|[X, Y ]t |  [X, X]t [Y, Y ]t
and
p
p
p
[X + Y, X + Y ]t  [X, X]t + [Y, Y ]t
(4.6.21)
(4.6.22)
Proof. Taking s = 0 and using |[X, Y ]t |  Var[0,t] ([X, Y ]) (4.6.21) follows from (4.6.17).
In turn using (4.6.21) we note that
[X + Y, X + Y ]t = [X, X]t + [Y, Y ]t + 2[X, Y ]t
p
 [X, X]t + [Y, Y ]t + 2 [X, X]t [Y, Y ]t
p
p
= ( [X, X]t + [Y, Y ]t )2 .
4.6. Quadratic Variation of Stochastic Integrators
125
Thus (4.6.22) follows.
The next inequality is a version of the Kunita-Watanabe inequality, that was proven in
the context of square integrable martingales.
Theorem 4.72. Let X, Y be stochastic integrators and let f, g be predictable processes.
Then for all T < 1
Z T
Z T
Z T
1
1
2
2
|fs gs |d|[X, Y ]|s  (
|fs | d[X, X]s ) (
|gs |2 d[Y, Y ]s ) 2 .
(4.6.23)
0
0
0
|At
[X, X]s ) 2 ([Y, Y ]t
|fs gs |d|[X, Y ]|s
=
k
X
j=0

k
X
j=0
k
X
j=0
Z
=(
T
0
Ar j )
|cj+1 dj+1 |([X, X]rj+1
c2j+1 ([X, X]rj+1
1
|fs |2 d[X, X]s ) 2 · (
D
(
|cj+1 dj+1 |(Arj+1
A
0
1
[Y, Y ]s ) 2
F
T
1
[X, X]rj )) 2 (
Z
1
[X, X]rj ) 2 ([Y, Y ]rj+1
R
and hence
Z
1
As |  ([X, X]t
T
Proof. Let us write At = |[X, Y ]|t = Var[0,t] ([X, Y ]). Note A0 = 0 by definition of quadratic
variation. We first observe that (4.6.23) holds for simple predictable processes f, g 2 S.
As seen in the proof of Theorem 4.29, we can assume that f, g are given by (4.3.11) and
(4.3.12). Using (4.6.15), it follows that for 0  s < t
k
X
1
[Y, Y ]rj ) 2
d2j+1 ([Y, Y ]rj+1
1
[Y, Y ]rj )) 2
j=0
T
0
1
|gs |2 d[Y, Y ]s ) 2 .
This proves (4.6.23) for f, g 2 S. Now using functional version of monotone class theorem,
Theorem 2.64 one can deduce that (4.6.23) continues to hold for all bounded predictable
processes f, g. Finally, for general f, g, the inequality follows by approximating f, g by
f n = f 1{|f |n} and g n = g 1{|g|n} respectively and using monotone convergence theorem
(recall, that integrals appearing in this result are Lebesgue-Stieltjes integrals with respect
to increasing processes.)
Remark 4.73. Equivalent Probability Measures continued: Let X be a stochastic integrator
on (⌦, F, P). Let Q be a probability measure equivalent to P. We have seen in Remark 4.15
126
Chapter 4. Stochastic Integration
that X is also a stochastic integrator on (⌦, F, Q) and the class L(X) under the two measures
R
is the same and for f 2 L(X), the stochastic integral f dX on the two spaces is identical.
It follows (directly from definition or from (4.6.1)) that the quadratic variation of X is the
same when X is considered on (⌦, F, P) or (⌦, F, Q).
4.7
Quadratic Variation of the Stochastic Integral
R
In this section, we will relate the quadratic variation of Y = f dX with the quadratic
variation of X. We will show that for stochastic integrators X, Z and f 2 L(X), g 2 L(Z)
Z
Z
Z
[ f dX, gdZ] = f gd[X, Z].
(4.7.1)
We begin with a simple result.
T
Lemma 4.74. Let X be a stochastic integrator, f 2 L(X), 0  u < 1, b be a Fu measurable
bounded random variable. Then
h = b1(u,1) f
F
is predictable, h 2 L(X) and
Z t
Z t
Z t
b1(u,1) f dX = b
1(u,1) f dX = b ( f dX
0
0
Z
0
u^t
f dX).
(4.7.2)
0
R
A
Proof. When f 2 S, validity of (4.7.2) can be verified directly as then h is also simple
predictable. Then, the class of f such that (4.7.2) is true can be seen to be closed under
bp-convergence and hence by Theorem 2.64, (4.7.2) is valid for all bounded predictable
processes. Finally, since b is bounded, say by c, |h|  c|f | and hence h 2 L(X). Now
(4.7.2) can be shown to be true for all f 2 L(X) by approximating f by f n = f 1{|f |n}
and using Dominated Convergence Theorem - Theorem 4.26.
D
Lemma 4.75. Let X, Z be a stochastic integrators, 0  u < 1, b be a Fu measurable
R
bounded random variable. Let g = b1(u,1) and Y = gdX. Then
Z t
Z t
Z t
Yt Zt =
Zs dYs +
Ys dZs +
gs d[X, Z]s .
(4.7.3)
0
0
Proof. Noting that Yt = b(Xt
Yt Zt =b (Zt Xt
0
Xu^t ) and writing Vt = Xt
Xu^t , we have,
Zt Xu^t )
=b (Xt Zt Xu^t Zu^t Xu^t Zt + Xu^t Zu^t )
Z t
Z u^t
Z t
Z
=b ( X dZ
X dZ +
Z dX
0
+ b ([X, Z]t
0
[X, Z]u^t
0
Xu^t (Zt
Zu^t )).
u^t
Z dX)
0
4.7. Quadratic Variation of the Stochastic Integral
Noting that
Z
t
X dZ
0
Z
127
u^t
X dZ
Xu^t (Zt
Zu^t ) =
0
R u^t
V
0
Z
t
V dZ
0
since Vs = 0 for 0  s  u,
dZ = 0 and we get
Z t
Z t
Z u^t
Yt Zt = b ( U dZ +
Z dX
Z dX + [X, Z]t
0
0
Invoking Lemma 4.74 and again using
R u^t
0
[X, Z]u^t ).
0
V dZ = 0 we conclude that (4.7.3) holds.
Theorem 4.76. Let X, Y be stochastic integrators and let f, h be bounded predictable
processes. Then
Z
Z
Z t
[ f dX, hdY ]t =
f hd[X, Y ].
(4.7.4)
0
Wt Z t = W0 Z 0 +
Z
t
f dX
W dZ +
0
Z
t
Z
F
Wt =
T
Proof. Fix a stochastic integrator Z and let K be the class of bounded predictable processes
f such that
Z
Z dW +
0
(4.7.5)
t
fs d[X, Z]s .
(4.7.6)
0
D
R
A
Easy to see that K is a linear space and that it is closed under bounded pointwise convergence of sequences. It trivially contains constants and we have seen in Lemma 4.75 that K
contains g = b1(u,1) , where 0  u < 1 and b is Fu measurable bounded random variable
and since S is the linear span of such processes, it follows that S ✓ K.
Now Theorem 2.64 implies that (4.7.5)-(4.7.6) hold for all bounded predictable processes. Comparing with (4.6.7), we conclude that for any stochastic integrator Z
Z
Z
[ f dX, Z] = f d[X, Z].
(4.7.7)
R
For Z = hdY , we can use (4.7.7) to conclude
Z
[Z, X] = hd[Y, X]
and using symmetry of the cross quadratic variation [X, Y ], we conclude
Z
[X, Z] = hd[X, Y ].
The two equations (4.7.7) - (4.7.8) together give
R
R
R
R
[ f dX, hdY ] = f d[X, hdY ]
R
= f hd[X, Y ].
(4.7.8)
(4.7.9)
128
Chapter 4. Stochastic Integration
We would like to show that (4.7.4) is true for all f 2 L(X) and h 2 L(Y ).
Theorem 4.77. Let X, Y be stochastic integrators and let f 2 L(X), g 2 L(Y ) and let
R
R
U = f dX, V = gdY . Then
Z t
[U, V ]t =
f gd[X, Y ].
(4.7.10)
0
R
Proof. Let us approximate f, g by f n = f 1{|f |n} and g n = g 1{|g|n} and let U n = f n dX,
R
ucp
ucp
V n = g n dY . Since f 2 L(X) and g 2 L(Y ), by definition U n ! U and V n ! V . Now
using Theorem 2.69 (also see the Remark following the result) we can get a subsequence
{nk } and a r.c.l.l. adapted increasing process H such that
k
k
|Utn | + |Vtn |  Ht 8k
1.
(4.7.11)
0
0
F
T
Using (4.7.4) for f n , g n (since they are bounded) we get invoking Theorem 4.29
Z t
Z t
n n
n n
n
n
U t V t = U 0 V0 +
Us dVs +
Vsn dUsn + [U n , V n ]t
0
0
(4.7.12)
Z t
Z t
Z t
n n
n n
n n
n n
= U 0 V0 +
Us gs dYs +
Vs fs dXs +
fs gs d[X, Y ]s .
0
A
Taking Y = X and g = f in (4.7.12) we get
Z t
Z t
n 2
n 2
n n
(Ut ) = (U0 ) + 2
Us fs dXs +
(fsn )2 d[X, X]s .
0
(4.7.13)
0
R
In (4.7.13), we would like to take limit as n ! 1. Since (fsn )2 = fs2 1{|f |n} increases to
fs2 , using monotone convergence theorem, we get
Z t
Z t
n 2
(fs ) d[X, X]s !
fs2 d[X, X]s .
(4.7.14)
0
0
D
For the stochastic integral term, taking limit along the subsequence {nk } (chosen so that
(4.7.11) holds) and using H f 2 L(X) (see Corollary 4.40) and dominated convergence
theorem (Theorem 4.26), we get
Z t
Z t
ucp
nk nk
Us fs dXs !
Us fs dXs .
(4.7.15)
0
0
ucp
Thus putting together (4.7.13), (4.7.14) and (4.7.15) along with U n ! U we conclude
Z t
Z t
2
2
(Ut ) = (U0 ) + 2
Us fs dXs +
(fs )2 d[X, X]s .
(4.7.16)
0
This implies
[U, U ]t =
0
Z
t
0
(fs )2 d[X, X]s .
(4.7.17)
4.7. Quadratic Variation of the Stochastic Integral
129
More importantly, this implies
Z t
(fs (!))2 d[X, X]s (!) < 1 8t < 1 a.s.
(4.7.18)
0
Likewise, we also have
Z
t
0
(gs (!))2 d[Y, Y ]s (!) < 1 8t < 1 a.s.
Now invoking Theorem 4.72 along with (4.7.18)-(4.7.19), we get
Z t
|fs (!)gs (!)|d|[X, Y ]|s (!) < 1 8t < 1 a.s.
(4.7.19)
(4.7.20)
0
0
T
and then using dominated convergence theorem (for signed measures) we conclude
Z t
Z t
n
n
fs (!)gs (!)d[X, Y ]s (!) !
fs (!)gs (!)d[X, Y ]s (!) 8t < 1 a.s.
(4.7.21)
0
F
In view of (4.7.21), taking limit in (4.7.12) along the subsequence {nk } and using argument
similar to the one leading to (4.7.16), we conclude
Z t
Z t
Z t
U t Vt = U 0 V0 +
Us gs dYs +
Vs fs dXs +
fs gs d[X, Y ]s
0
0
0
U t Vt = U 0 V0 +
Z
t
Us dVs +
0
R
and hence that
Z
A
which in turn implies
Vs dUs +
0
Z
t
fs gs d[X, Y ]s
0
t
fs gs d[X, Y ]s .
0
D
[U, V ]t =
Z
t
The earlier proof contains a proof of the following Theorem. Of course, this can also
be deduced by taking f = h and X = Y .
R
Theorem 4.78. Let X be stochastic integrators and let f 2 L(X) and let U = f dX.
Then
Z t
[U, U ]t =
f 2 d[X, X]
(4.7.22)
0
Remark 4.79. In Particular, it follows that for a stochastic integrator X, if f 2 L(X) then
Z t
fs2 d[X, X]s < 1 a.s. 8t < 1.
0
130
Chapter 4. Stochastic Integration
4.8
Ito’s Formula
Ito’s formula is a change of variable formula for stochastic integral. Let us look at the
familiar change of variable formula in usual Calculus. Let G be a continuously di↵erentiable
function on [0, 1) with derivative G0 = g and f be a continuously di↵erentiable function
on R. Then
Z
t
f (G(t)) = f (G(0)) +
f 0 (G(s))g(s)ds.
(4.8.1)
0
0+
T
d
This can be proven by observing that dt
f (G(t)) = f 0 (G(t))g(t) and using the fundamental
theorem of integral calculus. What can we say when G(t) is not continuously di↵erentiable?
Let us recast the change of variable formula as
Z t
f (G(t)) = f (G(0)) +
f 0 (G(s))dG(s).
(4.8.2)
h(") = sup{|f 0 (x1 )
f 0 (x2 )| :
F
Now this is true as long as G is a continuous function with finite variation. Let |G(s)|  K
for 0  s  t. Let
K  x1 , x2  K, |x1
G(t2 )| : 0  t1 , t2  t, |t1
t2 |  },
A
a( ) = sup{|G(t1 )
x2 |  "},
so that h(a( nt )) ! 0 as n ! 1 in view of uniform continuity of f on [ K, K] and G on
[0, t].
it
n.
Now using the mean value theorem, we get
R
Let us write tni =
f (G(tni+1 )) f (G(tni )) = f 0 (✓in )(G(tni+1 )
=[f 0 (G(tni )) + {f 0 (✓in )
D
where ✓in = G(tni ) + uni (G(tni+1 )
that
f (G(t))
G(tni ))
f 0 (G(tni ))}](G(tni+1 )
G(tni ))
(4.8.3)
G(tni )), for some uni , 0  uni  1. Now, it is easily seen
f (G(0)) =
n
X1
[f (G(tni+1 ))
f (G(tni ))]
(4.8.4)
i=0
and using dominated convergence theorem for Lebesgue-Stieltjes integration, we have
Z t
n
X1
lim
[f 0 (G(tni ))(G(tni+1 ) G(tni ))] =
f 0 (G(s))dG(s).
(4.8.5)
n!1
Since |G(tni+1 )
0+
i=0
G(tni )|  a(tn
1 ),
it follows that
|(f 0 (✓in )
f 0 (G(tni )))|  h(a(tn
1
))
(4.8.6)
4.8. Ito’s Formula
131
since ✓in = G(tni ) + uni (G(tni+1 )
|
n
X1
G(tni )), with uni , 0  uni  1. Hence,
[f 0 (✓in )
f 0 (G(tni ))](G(tni+1 )
G(tni ))|
i=0
 h(a(tn
1
 h(a(tn
1
))
n
X1
i=0
|G(tni+1 )
G(tni )|
(4.8.7)
))Var(0,t] (G)
! 0.
Now putting together (4.8.3)-(4.8.7) we conclude that (4.8.2) is true.
F
T
From the proof we see that unless the integrator has finite variation on [0, T ], the sum
of error terms may not go to zero. For a stochastic integrator, we have seen that the
quadratic variation is finite. This means we should keep track of first two terms in Taylor
expansion and take their limits (and prove that remainder goes to zero). Note that (4.8.3)
is essentially using Taylor expansion up to one term with remainder, but we had assumed
that f is only once continuously di↵erentiable.
A
The following lemma is a crucial step in the proof of the Ito formula. First part is
proven earlier, stated here for comparison and ease of reference.
R
Lemma 4.80. Let X, Y be stochastic integrators and let Z be an r.c.l.l. adapted process.
Let m # 0 and for m 1 let {⌧nm : n 1} be a m -partition for X, Y and Z. Then
1
X
and
D
n=0
1
X
n=0
Z⌧nm ^t (X
m ^t
Z⌧nm ^t (X⌧n+1
m ^t
⌧n+1
X⌧nm ^t )(Y
X⌧nm ^t )
ucp
!
Y⌧nm ^t )
m ^t
⌧n+1
Remark 4.81. Observe that if Z is continuous
Rt
Z
t
Z dX
(4.8.8)
Z
(4.8.9)
0
ucp
!
0 Z dX =
t
Z d[X, Y ].
0
Rt
0+ Z dX
Proof. The first part, (4.8.8) has been proved in Theorem 4.57. The second part for the
special case Z = 1 is proven in Theorem 4.62. For (4.8.9), note that
Ant = Btn
Ctn
Dtn
132
Chapter 4. Stochastic Integration
where
Ant =
Btn =
Ctn =
Dtn =
1
X
n=0
1
X
n=0
1
X
n=0
1
X
n=0
m ^t
Z⌧nm ^t (X⌧n+1
m ^t
X⌧nm ^t )(Y⌧n+1
m ^t Y⌧ m ^t
Z⌧nm ^t (X⌧n+1
n+1
Y⌧nm ^t ),
X⌧nm ^t Y⌧nm ^t ),
m ^t
Z⌧nm ^t X⌧nm ^t (Y⌧n+1
Y⌧nm ^t ),
m ^t
Z⌧nm ^t Y⌧nm ^t (X⌧n+1
X⌧nm ^t ).
0
0
F
T
Now using (4.8.8), we have
Z t
ucp
Btn !
Z d(XY )
0
Z t
Z t
Z
n ucp
Ct !
Z X dY =
Z dS where S = X dY
0
0
Z t
Z t
Z
n ucp
Dt !
Z Y dX =
Z dR where R = Y dX.
0
X dY
R
where Vt = Xt Yt X0 Y0
This completes the proof.
Rt
A
Here we have used Theorem 4.29. Using bilinearity of integration, it follows that
Z t
n ucp
At !
Z dV
Remark 4.82. If {⌧nm : m
Rt
0
0
Y dX. As seen in Theorem 4.62, Vt = [X, Y ]t .
1} is a sequence of partitions of [0, 1) via stop times
D
0 = ⌧0m < ⌧1m < ⌧2m . . . ; ⌧nm " 1, m
1
such that for all n, m
m
(⌧n+1
⌧nm )  2
m
then (4.8.9) holds for all r.c.l.l. adapted processes X, Y, Z.
We will first prove a single variable change of variable formula for a continuous stochastic
integrator and then go on to the multivariate version.
Now let us fix a twice continuously di↵erentiable function f . The standard version of
Taylor’s theorem gives
Z b
0
f (b) = f (a) + f (a)(b a) +
f 00 (s)(b s)ds.
(4.8.10)
a
4.8. Ito’s Formula
133
However, we need the expansion up to two terms with an estimate on the remainder. Let
us write
1
f (b) = f (a) + f 0 (a)(b a) + f 00 (a)(b a)2 + Rf (a, b)
(4.8.11)
2
where
Z
b
Rf (a, b) =
[f 00 (s)
f 00 (a)](b
s)ds.
a
Since f 00 is assumed to be continuous, for any K < 1, f 00 is uniformly continuous and
bounded on [ K, K] so that
lim ⇤f (K, ) = 0
!0
where
⇤f (K, ) = sup{|Rf (a, b)(b
a)
2
| : a, b 2 [ K, K], 0 < |b
a| < }.
T
Here is the univariate version of the Ito formula for continuous stochastic integrators.
F
Theorem 4.83. (Ito’s formula - first version) Let f be a a twice continuously di↵erentiable function on R and X be a continuous stochastic integrator. Then
Z
Z t
1 t 00
0
f (Xt ) = f (X0 ) +
f (Xu )1{u>0} dXu +
f (Xu )d[X, X]u .
2 0
0
Proof. Fix t. Let tni =
ti
n
for i
A
Remark 4.84. Equivalently, we can write the formula as
Z t
Z
1 t 00
f (Xt ) = f (X0 ) +
f 0 (Xu )dXu +
f (Xu )d[X, X]u .
2 0
0+
0, n
1. Let
D
R
Uin = f (Xtni+1 )
f (Xtni ),
(4.8.12)
Vin = f 0 (Xtni )(Xtni+1 Xtni ),
1
Win = f 00 (Xtni )(Xtni+1 Xtni )2 ,
2
n
Ri = Rf (Xtni , Xtni+1 ).
(4.8.13)
(4.8.14)
(4.8.15)
Then one has
Uin = Vin + Win + Rin (Xtni+1
n
X1
Uin = f (Xt )
Xtni )2 ,
f (X0 ).
(4.8.16)
(4.8.17)
i=0
Further using (4.8.8) and (4.8.9) (see remark 4.81), we get
Z t
n
X1
n
Vi !
f 0 (Xu )1{u>0} dXu in probability,
i=0
0
(4.8.18)
134
Chapter 4. Stochastic Integration
n
X1
i=0
Win !
1
2
Z
t
f 00 (Xu )d[X, X]u in probability.
(4.8.19)
0
In view of (4.8.12)-(4.8.19) and the fact that
n
X1
Xtni )2 ! [X, X]t in probability
(Xtni+1
i=0
it suffices to prove that,
sup Rin ! 0 in probability.
i
Indeed, to complete the proof we will show
sup Rin ! 0 almost surely.
(4.8.20)
i
T
Observe that
|Rin (!)|  |Rf (Xtni (!), Xtni+1 (!))|.
n (!)
F
For each !, u 7! Xu (!) is uniformly continuous on [0, t] and hence
= [sup|Xtni+1 (!)
i
Let Kt (!) = sup0ut |Xu (!)| Now
Xtni (!)| ] ! 0.
i
and since
n (!)
A
sup|Rin (!)|  ⇤f (Kt (!),
n (!))
! 0, it follows that (4.8.20) is valid completing the proof.
R
Applying the Ito formula with f (x) = xm , m 2, we get
Z t
Z
m(m 1) t m
Xtm = X0m +
mXum 1 1{u>0} dXu +
Xu
2
0
0
2
1{u>0} d[X, X]u
D
and taking f (x) = exp(x) we get
Z
t
1
exp(Xt ) = exp(X0 ) +
exp(Xu )dXu +
2
0+
Z
t
exp(Xu )d[X, X]u .
0
We now turn to the multidimensional version of the Ito formula. Its proof given below
is in the same spirit as the one given above in the one dimensional case and is based on
the Taylor series expansion of a function. This idea is classical and the proof given here is
a simplification of the proof presented in Metivier [48]. A similar proof was also given in
Kallianpur and Karandikar [31].
We will first prove the required version of Taylor’s theorem. Here, | · | denotes the
Euclidean norm on Rd , U will denote a fixed open convex subset of Rd and C 1,2 ([0, 1) ⇥ U )
4.8. Ito’s Formula
135
denotes the class of functions f : [0, 1) ⇥ U 7! R that are once continuously di↵erentiable
in t and twice continuously di↵erentiable in x 2 U . Also, for f 2 C 1,2 ([0, 1) ⇥ U), f0
denotes the partial derivative of f in the t variable, fj denotes the partial derivative of f
w.r.t. j th coordinate of x = (x1 , . . . , xd ) and fjk denote the partial derivative of fj w.r.t.
k th coordinate of x = (x1 , . . . , xd ).
Lemma 4.85. Let f 2 C 1,2 ([0, 1) ⇥ U ). Define h : [0, 1) ⇥ U ⇥ U ! R as follows. For
t 2 [0, 1), y = (y 1 , . . . , y d ), x = (x1 , . . . , xd ) 2 U , let
h(t, y, x) =f (t, y)
d
X
f (t, x)
(y j
xj )fj (t, x)
j=1
(y
j
j
x )(y
(4.8.21)
k
k
x )fjk (t, x).
T
1
2
d
X
j,k=1
F
Then there exist continuous functions gjk : [0, 1) ⇥ U ⇥ U ! R such that for t 2 [0, 1),
y = (y 1 , . . . , y d ), x = (x1 , . . . , xd ) 2 U ,
xj )(y k
xk ).
Further, the following holds. Define for T < 1, K ✓ U and
> 0,
h(t, y, x) =
d
X
gjk (t, y, x)(y j
A
j,k=1
(T, K, ) = sup{|h(t, y, x)||y
x|
2
: 0  t  T, x 2 K, y 2 K, 0 < |x
(4.8.22)
y|  }
R
and
⇤(T, K) = sup{|h(t, y, x)||y
x|
2
: 0  t  T, x 2 K, y 2 K, x 6= y}.
and
D
Then for T < 1 and a compact subset K ✓ U , we have
⇤(T, K) < 1
lim (T, K, ) = 0.
#0
Proof. Fix t 2 [0, 1). For 0  s  1, define
g(s, y, x) = f (t, x + s(y
x))
f (t, x)
s
d
X
(y j
xj )fj (t, x)
j=1
s2
2
d
X
j,k=1
(y j
xj )(y k
xk )fjk (t, x)
136
Chapter 4. Stochastic Integration
where x = (x1 , . . . , xd ), y = (y 1 , . . . , y d ) 2 Rd . Then g(0, y, x) = 0 and g(1, y, x) =
d
d2
00
0
h(t, y, x). Writing ds
g = g 0 and ds
2 g = g , we can check that g (0, y, x) = 0 and
g 00 (s, y, x) =
d
X
(fjk (t, x + s(y
fjk (t, x))(y j
x))
xj )(y k
xk ).
j,k=1
Noting that g(0, y, x) = g 0 (0, y, x) = 0, by Taylor’s theorem (see remainder form given in
equation (4.8.11)) we have
h(t, y, x) =g(1, y, x)
Z 1
=
(1 s)g 00 (s, y, x)ds.
(4.8.23)
Thus (4.8.22) is satisfied where {gjk } are defined by
Z 1
gjk (t, y, x) =
(1 s)(fjk (t, x + s(y
0
T
0
x))
fjk (t, x))ds.
(4.8.24)
F
The desired estimates on h follow from (4.8.22) and (4.8.24).
A
Theorem 4.86. (Ito’s Formula for Continuous Stochastic Integrators) Let f 2
C 1,2 ([0, 1) ⇥ U ) where U ✓ Rd is a convex open set and let Xt = (Xt1 , . . . , Xtd ) be an
U -valued continuous process where each X j is a stochastic integrator. Then
Z t
d Z t
X
f (t, Xt ) =f (0, X0 ) +
f0 (s, Xs )ds +
fj (s, Xs )dXsj
0
d
d
R
1X X
+
2
j=1 k=1
Z
t
j=1
0+
(4.8.25)
fjk (s, Xs )d[X j , X k ]s .
0
D
Proof. Suffices to prove the equality (4.8.25) for a fixed t a.s. since both sides are continuous
processes. Once again let tni = ti
n and
Vin =
d
X
fj (tni , Xtni )(Xtjn
i+1
j=1
Win =
d
1 X
fjk (tni , Xtni )(Xtjn
i+1
2
j,k=1
Rin
Xtjn )
i
Xtjn )(Xtkni+1
i
Xtkni )
= h(tni , Xtni , Xtni+1 )
From the definition of h - (4.8.21) it follows that
f (tni , Xtni+1 )
f (tni , Xtni ) = Vin + Win + Rin
(4.8.26)
4.8. Ito’s Formula
137
Now
f (t, Xt )
f (0, X0 ) =
=
n
X1
i=0
n
X1
(f (tni+1 , Xtni+1 )
f (tni , Xtni ))
(f (tni+1 , Xtni+1 )
f (tni , Xtni+1 ))
i=0
+
n
X1
(f (tni , Xtni+1 )
f (tni , Xtni ))
i=0
=
n
X1
(Uin + Vin + Win + Rin )
i=0
in view of (4.8.26), where
T
Uin =f (tni+1 , Xtni+1 ) f (tni , Xtni+1 )
Z tn
i+1
=
f0 (u, Xtni+1 )du
tn
i
(4.8.27)
F
and hence, by the dominated convergence theorem for Lebesgue integrals, we have
Z t
n
X1
n
Ui !
f0 (s, Xs )ds.
(4.8.28)
0
A
i=0
Using (4.8.8) and (4.8.9), we get (see remark 4.81)
n
d Z t
X1
X
n
Vi !
fj (s, Xs )dXsj in probability
and
n
X1
d Z
1X t
!
fjk (s, Xs )d[X j , X k ]s in probability.
2
0
D
i=0
Win
j=1
(4.8.29)
0+
R
i=0
(4.8.30)
j,k=1
In view of these observations, the result, namely (4.8.25) would follow once we show that
n
X1
i=0
Rin ! 0 in probability.
Now let K t,! = {Xs (!) : 0  s  t} and n (!) = supi |(Xtni+1 (!)
compact, n (!) ! 0 and hence by Lemma 4.85 for every ! 2 ⌦
(K t,! ,
n (!))
! 0.
(!)
n (!))(Xtn
i+1
Xtni (!))|. Then K t,! is
(4.8.32)
Since
|Rin |  (K t,! ,
(4.8.31)
Xtni (!))2
138
we have
Chapter 4. Stochastic Integration
n
X1
i=0
|Rin |  (K t,! ,
n (!))
n
X1
(Xtni+1 (!)
Xtni (!))2 .
i=0
The first factor above converges to 0 pointwise as seen in (4.8.32) and the second factor
converges to [X, X]t in probability, we conclude that (4.8.31) is true completing the proof
as noted earlier.
Theorem 4.87. (Ito’s Formula for r.c.l.l. Stochastic Integrators) Let U be a convex
open subset of Rd . Let f 2 C 1,2 ([0, 1) ⇥ U ). Let X 1 , . . . , X d be stochastic integrators
Xt := (Xt1 , . . . , Xtd ) is U -valued. Further, suppose X is also U -valued. Then
Z t
d Z t
X
f (t, Xt ) =f (0, X0 ) +
f0 (s, Xs )ds +
fj (s, Xs )dXsj
d
j=1 k=1
+
X
t
fjk (s, Xs )d[X j , X k ]s
0
{f (s, Xs )
d X
d
X
d
X
(4.8.33)
f (s, Xs )
fj (s, Xs ) Xsj
j=1
1
fjk (s, Xs )( Xsj )( Xsk )}.
2
A
0<st
Z
0+
F
d
1X X
+
2
j=1
T
0
j=1 k=1
D
R
Proof. Let us begin by examining the last term. Firstly, what appears to be a sum of
uncountably many terms is really a sum of countable number of terms- since the summand
is zero whenever Xs = Xs . Also, the last term equals
X
Dt =
h(s, Xs , Xs )
(4.8.34)
0<st
where h is defined by (4.8.21). By Lemma 4.85 for 0 < s  t
|h(s, Xs (!), Xs (!))|  ⇤(K t,! )| Xs (!)|2
where
K t,! = {Xt (!) : 0  t  T } [ {Xt (!) : 0  t  T }
(4.8.35)
Here K t,! is compact (see Exercise 2.1) and thus by Lemma 4.85, ⇤(K t,! ) < 1. As a
consequence, we have
X
X
|h(s, Xs (!), Xs (!))|  ⇤(K t,! )
( Xs (!))2
0<st
0<st
4.8. Ito’s Formula
139
and invoking Corollary 4.65 we conclude that series in (4.8.34) converges absolutely, a.s.
The rest of the argument is on the lines of the proof in the case of continuous stochastic
integrators except that this time the remainder term after expansion up to two terms does
not go to zero yielding an additional term. Also, the proof given below requires use of
partitions via stop times.
For each n
1, define a sequence {⌧in : i
⌧0n = 0 and for i 0,
n
⌧i+1
= inf{t > ⌧in : max{|Xt
1} of stop times inductively as follows:
X⌧in |, |Xt
X⌧in |, |t
⌧in |}
2
n
}
(4.8.36)
Let us note that each ⌧in is a stop time (see Theorem 2.44),
n
0 = ⌧0n < ⌧1n < . . . < ⌧m
< ...,
T
n
1, ⌧m
" 1 as m ! 1
8n
and
n : m
Thus, {⌧m
0}, n
⌧in )  2
n
.
F
n
(⌧i+1
1 satisfies the conditions of Lemma 4.80.
A
Fix t > 0. On the lines of the proof in the continuous case let
n
n ^t )
Ani = f (⌧i+1
^ t, X⌧i+1
=
d
X
j=1
Win
fj (⌧in ^ t, X⌧in ^t )(X⌧j n
i+1 ^t
R
Vin
n ^t )
f (⌧in ^ t, X⌧i+1
i
d
1 X
=
fjk (⌧in ^ t, X⌧in ^t )(X⌧j n ^t
i+1
2
n ^t
X⌧j n ^t )(X⌧ki+1
i
X⌧kin ^t )
D
j,k=1
Rin
Then
X⌧j n ^t )
n ^t , X⌧ n ^t )
= h(⌧in ^ t, X⌧i+1
i
n
n ^t )
f (⌧i+1
^ t, X⌧i+1
f (⌧in ^ t, X⌧in ^t ) = Ani + Vin + Win + Rin
and hence
f (t, Xt )
f (0, X0 ) =
1
X
(Ani + Vin + Win + Rin ).
i=0
Now,
1
X
i=0
Ani
=
1 Z
X
i=0
n ^t
⌧i+1
⌧in ^t
n ^t )ds
f0 (s, X⌧i+1
(4.8.37)
140
Chapter 4. Stochastic Integration
and hence for all !
1
X
Ani (!)
i=0
Z
!
t
f0 (s, Xs (!))ds
0
Since for every !, Xs (!) = Xs (!) for all but countably many s, we can conclude
Z t
1
X
n
Ai !
f0 (s, Xs )ds in probability.
(4.8.38)
0
i=0
By Lemma 4.80 and the remark following it,
1
d Z t
X
X
Vin !
fj (s, Xs )dXsj
and
1
X
j=1
d
Win
i=0
d
1 XX
!
2
j=1 k=1
Z
t
fjk (s, Xs )d[X j , X k ]
T
i=0
(4.8.39)
0+
0
(4.8.40)
1
X
i=0
n ^t , X⌧ n ^t ) !
h(⌧in ^ t, X⌧i+1
i
A
Rn =
F
n ^t , X⌧ n ^t ), to complete
in probability. In view of (4.8.37)-(4.8.40), and Rin = h(⌧in ^ t, X⌧i+1
i
the proof of the result it suffices to prove
Let us partition indices i into three sets:
large as follows:
h(s, Xs , Xs ).
(4.8.41)
0<st
n ^t , X⌧ n ^t ) is zero, is small and is
^ t, X⌧i+1
i
n
0 : ⌧i+1
(!) ^ t = ⌧in (!) ^ t}
R
H n (!) ={i
h(⌧in
X
n ^t (!)|  2 · 2
E n (!) ={i 62 H n : | X⌧i+1
n ^t (!)| > 2 · 2
F n (!) ={i 62 H n : | X⌧i+1
n
n
}
}.
D
n ^t (!), X⌧ n ^t (!))| = 0 and thus writing
For i 2 H n (!), |h(⌧in (!) ^ t, X⌧i+1
i
X
n ^t (!), X⌧ n ^t (!)),
B n (!) =
h(⌧in (!) ^ t, X⌧i+1
i
i2E n (!)
C n (!) =
X
i2F n (!)
n ^t (!), X⌧ n ^t (!))
h(⌧in (!) ^ t, X⌧i+1
i
we observe that
Rn = B n + C n .
n ), then |X
Note that for any j if u, v 2 (⌧jn , ⌧j+1
Xv |  2.2
u
n
n
n ), |X
2 distance from X⌧jn . As a result, for any v 2 (⌧j , ⌧j+1
v
n
as Xu , Xv are within
Xv |  2.2 n . Thus, if
4.8. Ito’s Formula
141
| Xs (!)| > 2·2 n for s 2 (0, t], then s must equal ⌧jn (!) ^ t for some j with i = j 1 2 F n .
i.e.
n
if s 2 (0, t] and | Xs (!)| > 2 · 2 n then s = ⌧i+1
^ t for i 2 F n (!).
(4.8.42)
Hence for i 2 E n (!),
n ^t (!)
|X⌧i+1
n ^t (!)
X⌧in ^t (!)| |(X )⌧i+1
3 · 2
and hence
|B n (!)| 
X
i2E n (!)
n ^t (!)|
X⌧in ^t (!)| + | X⌧i+1
n
n ^t (!), X⌧ n ^t (!))|
|h(⌧in (!) ^ t, X⌧i+1
i
 (K t,! , 3 · 2
n
)
X
i
X⌧in ^t (!)|2
n ^t (!)
|X⌧i+1
P
n
where
defined by (4.8.35) is compact. Since
X⌧in ^t |2 converges to
i |X⌧i+1 ^t
P
j
j
t,! , 3 · 2 n ) ! 0 for all !, invoking Lemma 4.85 it
j [X , X ]t in probability and (K
follows that
B n ! 0 in probability.
F
T
K t,!
Thus to complete the proof of (4.8.41), it would suffice to show that
X
Cn !
h(s, Xs , Xs ) in probability.
A
(4.8.43)
0<st
R
Let G(!) = {s 2 (0, t] : | Xs (!)| > 0}. Since X is an r.c.l.l. process, G(!) is a countable
set for every !. Fix ! and for s 2 G(!), define
X
n (!)^t , X⌧ n ^t (!))1{⌧ n (!)^t=s} .
ans (!) :=
h(⌧in (!) ^ t, X⌧i+1
i
i+1
i2F n (!)
Then
X
D
C n (!) =
i2F n (!)
n,
If | Xs (!)| > 2 · 2
then
seen in (4.8.42)) and hence
n ^t (!), X⌧ n ^t (!)) =
h(⌧in ^ t(!), X⌧i+1
i
ans (!)
=
for all
ans (!).
s2G(!)
h(⌧in (!) ^ t, Xs (!), X⌧in ^t (!))
ans (!) ! h(s, Xs (!), Xs (!))
X
n (!) ^ t (as
with s = ⌧i+1
!.
For i 2 F n (!),
n ^t (!)
|X⌧i+1
n ^t (!)
X⌧in ^t (!)| |(X )⌧i+1
2
n
n ^t (!)|
X⌧in ^t (!)| + | X⌧i+1
n ^t (!)|
+ | X⌧i+1
n ^t (!)|
2| X⌧i+1
(4.8.44)
142
Chapter 4. Stochastic Integration
n (!) ^ t for some i 2 F n (!) and then
Thus if |ans (!)| =
6 0, then s = ⌧i+1
n ^t (!)
|ans (!)| ⇤(K t,! )|X⌧i+1
X⌧in ^t (!)|2
2
n ^t (!)|
4⇤(K t,! )| X⌧i+1
(4.8.45)
=4⇤(K t,! )| Xs (!)|2 .
Let Cs (!) = 4⇤(K t,! )| Xs (!)|2 . Then
X
X
Cs (!) = 4⇤(K t,! )
| Xs (!)|2
s
and hence by Corollary 4.65
s
X
s
Cs (!) < 1
a.s.
T
Using Weierstrass’s M -test (series version of the dominated convergence theorem) along
with |ans (!)|  Cs (!) (recall (4.8.44)), we get
X
X
C n (!) =
ans (!) !
h(s, Xs (!), Xs (!)) a.s.
(4.8.46)
s2G(!)
F
s2G(!)
We have proved that the convergence in (4.8.43) holds almost surely and hence in probability. This completes the proof.
A
Corollary 4.88. Let f , X be as in Theorem 4.87. Then Zt = f (t, Xt ) is a stochastic
integrator.
R
Proof. In the Ito formula (4.8.33) that expresses f (t, Xt ), it is clear that the terms involving
integral are stochastic integrators. We had seen that the last term is
X
Dt =
h(s, Xs Xs )
D
0<st
h is defined by (4.8.21). By Lemma 4.85
|h(s, Xs (!), Xs (!))|  ⇤(K t,! )| Xs (!)|2
for 0 < s  t, where
K t,! = {Xs (!) : 0  s  t} [ {Xs (!) : 0  s  t}.
Hence
Var[0,T ] (D)(!)  ⇤(K T,! )[X, X]T .
Thus D is a process with finite variation and hence is a stochastic integrator, completing
the proof.
4.8. Ito’s Formula
143
We have seen earlier in Lemma 4.65 that [X, Y ]t = ( X)t ( Y )t . Thus carefully
examining the right hand side of the Ito formula (4.8.33), we see that we are adding and
subtracting a term
d X
d
X
1
fjk (s, Xs )( Xsj )( Xsk ).
2
j=1 k=1
Let us introduce - for now in an ad-hoc manner [X, Y ](c) for semimartingales X, Y :
X
(c)
[X, Y ]t = [X, Y ]t
( X)s ( Y )s .
(4.8.47)
0<st
Later we will show that this is the cross-quadratic variation of continuous martingale parts
of X and Y (See Theorem 8.76). For now we observe the following.
F
T
Lemma 4.89. Let X, Y be semimartingales and h be a locally bounded predictable process.
Then [X, Y ](c) is a continuous process and further
Z
Z
X
(c)
hd[X, Y ] = hd[X, Y ]
hs ( X)s ( Y )s .
(4.8.48)
0<st
R
A
Proof. Continuity of [X, Y ](c) follows from its definition (4.89) and part (iii) of Lemma
4.65. The identity (4.8.48) for simple functions h 2 S follows by direct verification and
hence follows for all bounded predictable processes by monotone class theorem.
Using Lemma 4.89, we can recast the Ito formula in an alternate form
D
Theorem 4.90. Ito’s formula for r.c.l.l. stochastic integrators
Let U be a convex open subset of Rd . Let f 2 C 1,2 ([0, 1)⇥U ). Let X 1 , . . . , X d be stochastic
integrators Xt := (Xt1 , . . . , Xtd ) is U -valued. Further, suppose X is also U -valued. Then
Z t
d Z t
X
f (t, Xt ) =f (0, X0 ) +
f0 (s, Xs )ds +
fj (s, Xs )dXsj
0
d
+
d
1X X
2
j=1 k=1
+
X
0<st
Z
j=1
t
0+
fjk (s, Xs )d[X j , X k ](c)
0+
{f (s, Xs )
f (s, Xs )
d
X
j=1
fj (s, Xs ) Xsj }.
(4.8.49)
144
Chapter 4. Stochastic Integration
Exercise 4.91. Let S be a (0, 1)-valued continuous stochastic integrator. Show that Rt =
(St ) 1 is also a stochastic integrator and
Z t
Z t
2
Rt = R0
(Ss ) dSs +
(Ss ) 3 d[S, S]s .
0+
0+
Exercise 4.92. Let S be a (0, 1)-valued r.c.l.l. stochastic integrator with St > 0 for t > 0.
Show that Rt = (St ) 1 is also a stochastic integrator and
Z t
Z t
X
2
Rt = R0
(Ss ) dSs +
(Ss ) 3 d[S, S](c)
+
u(Ss , Ss )
s
0+
0+
where u(y, x) =
1
y
1
x
+ (y
0<st
x) x12 .
(i) Show that S satisfies
t
Su dXu .
1
2 [X, X]t ).
(4.8.50)
0+
F
St = exp(X0 ) +
Z
T
Exercise 4.93. Let X be a continuous stochastic integrator and let St = exp(Xt
Hint: Apply Ito formula for f (X, [X, X]) for suitable function f .
A
(ii) Let Z satisfy
Zt = exp(X0 ) +
Z
t
Zu dXu .
(4.8.51)
0+
R
Show that Z = S.
Hint: Let Yt = Zt exp( Xt + 12 [X, X]t ) and applying Ito formula, conclude that Yt =
Y0 = 1.
D
(iii) Show that
(iv) Let Rt = (St )
1.
Xt = X0 +
Show that R satisfies
Z t
1
Rt = (S0 )
(Su )
0+
1
Z
t
0+
Su 1 dSu .
dXu +
Z
which in turn is same as (see also Exercise 4.91)
Z t
Z
1
2
Rt = (S0 )
(Su ) dSu +
0+
Hint: Use part (i) above and Rt = exp(Yt
t
(Su )
1
d[X, X]u
(4.8.52)
(Su )
3
d[S, S]u .
(4.8.53)
0+
t
0+
1
2 [Y, Y ]t )
where Yt =
Xt + [X, X]t .
4.9. The Emery Topology
145
Exercise 4.94. Let X, Y be a (0, 1)-valued continuous stochastic integrators and let U, V
be solutions to
Z t
Ut = exp(X0 ) +
Uu dXu
(4.8.54)
0+
and
Vt = exp(Y0 ) +
Z
t
Vu dYu .
Let Wt = Ut Vt . Show that W is the unique solution to
Z t
Z t
Z
Wt = exp(X0 Y0 ) +
Wu dXu +
Wu dYu +
0+
4.9
(4.8.55)
0+
0+
t
Wu d[X, Y ]u .
(4.8.56)
0+
The Emery Topology
T
On the class of stochastic integrators, we define a natural metric dem as follows. Let S1
be the class of simple predictable processes f such that |f |  1. For stochastic integrators
X, Y , let
Z
Z
f dX,
f dY ) : f 2 S1 }
F
dem (X, Y ) = sup{ducp (
(4.9.1)
R
A
Easy to see that dem is a metric on the class of stochastic integrators. This metric was
defined by Emery [16] and the induced topology is called the Emery topology. If X n
em
converges to X in dem metric, we will write it as X n ! X. Taking f = 1 in (4.9.1), it
follows that
ducp (X, Y )  dem (X, Y )
(4.9.2)
and thus convergence in Emery topology implies convergence in ducp metric. If A is an
increasing process, then it is easy to see that dem (A, 0)  ducp (A, 0) and hence we have
D
dem (A, 0) = ducp (A, 0).
(4.9.3)
Let us define a metric dvar on V as follows: for B, C 2 V
dvar (B, C) = ducp (Var(B
C), 0).
Lemma 4.95. Let B, C 2 V. Then
dem (B, C)  dvar (B, C).
Proof. Note that for any predictable f with |f |  1
Z T
Z T
|
f dB
f dC|  Var[0,T ] (B
0
0
The result follows from this observation.
C).
(4.9.4)
146
Chapter 4. Stochastic Integration
As a consequence of (2.5.2)-(2.5.3), it can be seen that dem (X n , X) ! 0 if and only if
for all T > 0, > 0
Z t
Z t
lim [ sup P( sup | f dX n
f dX| > )] = 0
(4.9.5)
n!1 f :2S
1
and likewise that
0tT
0
0
{X n }
is Cauchy in dem is and only if for all T > 0, > 0
Z t
Z t
n
lim [ sup P( sup | f dX
f dX k | > )] = 0.
n,k!1 f :2S1
0tT
0
(4.9.6)
0
Theorem 4.96. For a bounded predictable process f with |f |  1 one has
Z
Z
ducp ( f dX, f dY )  dem (X, Y ).
(4.9.7)
A
F
T
Proof. Let K1 be the class of bounded predictable processes for which (4.9.7) is true. Then
using the dominated convergence theorem and the definition of ducp , it follows that K1 is
closed under bp-convergence and contains S1 .
Let K be the class of bounded predictable processes g such that g̃ = max(min(g, 1), 1) 2
K1 . Then it is easy to see that K is closed under bp-convergence and contains S. Thus K
is the class of all bounded predictable processes g and hence K1 contains all predictable
processes bounded by 1.
R
Corollary 4.97. If Xn , X are semimartingales such that dem (X n , X) ! 0 then for all
T > 0, > 0
Z t
Z t
lim [ sup P( sup | f dX n
f dX| > )] = 0
(4.9.8)
n!1 f :2K
1
0tT
0
0
where K1 is the class of predictable processes bounded by 1.
D
Linearity of the stochastic integral and the definition of the metric dem yields the
inequality
dem (U + V, X + Y )  dem (U, X) + dem (V, Y )
em
em
em
which in turn implies that if X n ! X and Y n ! Y then (X n + Y n ) ! (X + Y ).
We will now prove an important property of the metric dem .
Theorem 4.98. The space of stochastic integrators is complete under the metric dem .
Proof. Let {X n } be a Cauchy sequence in dem metric. Then by (4.9.2) it is also cauchy
in ducp metric and so by Theorem 2.68 there exists an r.c.l.l. adapted process X such that
ucp
em
X n ! X. We will show that X is a stochastic integrator and X n ! X.
4.9. The Emery Topology
147
Let an = supk 1 dem (X n , X n+k ). Then an ! 0 since X n is cauchy in dem . For
R
e P), consider Y n (f ) = f dX n . Then (in view of (4.9.7)) for bounded predictable
f 2 B(⌦,
f with |f |  c, we have
ducp (Y n (f ), Y n+k (f ))  c an , for all f, k
1
(4.9.9)
and hence for bounded predictable processes f , {Y n (f )} is Cauchy in ducp and hence by
Theorem 2.68, Y n (f ) converges to say Y (f ) 2 R0 (⌦, (F⇧ ), P) and
ducp (Y n (f ), Y (f ))  can , for all predictable f, |f |  c.
(4.9.10)
F
T
For simple predictable f , we can directly verify that Y (f ) = JX (f ). We will show
ucp
bp
(using the standard 3" ) argument that Y (f m ) ! Y (f ) if f m ! f and {f m } are uniformly bounded. Y (f ) would then be the required extension in the definition of stochastic
bp
integrator proving that X is a stochastic integrator. Let us fix f m ! f where {f m } are
uniformly bounded. Dividing by the uniform upper bound if necessary, we can assume that
|f m |  1. We wish to show that ducp (Y (f m ), Y (f )) ! 0 as m ! 1.
Given " > 0, first choose and fix n⇤ such that an⇤ < 3" . Then
⇤
⇤
⇤
ducp (Y (f m ), Y (f ))  ducp (Y (f m ), Y n (f m )) + ducp (Y n (f m ), Y n (f ))
⇤
+ ducp (Y n (f ), Y (f ))
D
R
A
(4.9.11)
⇤
⇤
 an⇤ + ducp (Y n (f m ), Y n (f )) + an⇤
"
"
⇤
⇤
 + ducp (Y n (f m ), Y n (f )) + .
3
3
R m
⇤ ucp R
⇤
bp
m
m
n
Since f
! f and {f } is bounded by 1, f dX
! f dX n and hence we can
choose m⇤ (depends upon n⇤ which has been chosen and fixed earlier) such that for m m⇤
one has
"
⇤
⇤
ducp (Y n (f m ), Y n (f )) 
3
and hence using (4.9.11) we get for m m⇤ ,
ducp (Y (f m ), Y (f ))  ".
R
Thus, Y (f ) is the required extension of { f dX : f 2 S} proving that X is a stochastic
integrator and
Z
Y (f ) =
Recalling that Y n (f ) =
R
f dX.
f dX n , the equation (4.9.10) implies
dem (X n , X)  an
with an ! 0. This completes the proof of completeness of dem .
148
Chapter 4. Stochastic Integration
We will now strengthen the conclusion in Theorem 4.46 by showing that the convergence
is actually in the Emery topology.
Theorem 4.99. Suppose Y n , Y 2 R0 (⌦, (F⇧ ), P), Y n
grator. Then
Z
Z
em
(Y n ) dX
!
ucp
! Y and X is a stochastic inte-
Y dX.
R
R
Proof. As noted in Thoerem 4.29, (Y n ) dX and Y dX are stochastic integrators.
Further, we need to show that we need to show that for T < 1, > 0
Z t
Z t
n
lim [ sup P( sup | (Y ) f dX
Y f dX| > )] = 0.
(4.9.12)
n!1 f :2S
1
0tT
0
0
T
We will prove this by contradiction. Suppose (4.9.12) is not true. Then there exists an "
and a subsequence {nk } such that
Z t
Z t
nk
sup P( sup | (Y ) f dX
Y f dX| > ) " 8k 1.
(4.9.13)
0tT
0
0
2 S1 such that
Z t
k
P( sup | (Y n ) f k dX
F
For each k get
fk
0tT
0
ucp
k
Z
t
Y f k dX| > )
".
(4.9.14)
0
A
f :2S1
R
Now let g k = (Y n
Y ) f k . Since Y n ! Y and f k are uniformly bounded by 1, it
ucp
follows that g k ! 0 and hence by Theorem 4.46,
Z t
Z t
nk
k
P( sup | (Y ) f dX
Y f k dX| > ) ! 0.
0tT
0
0
D
This contradicts (4.9.14). This proves (4.9.12).
We will first prove a lemma and then go on to the result mentioned above.
Lemma 4.100. Suppose U n 2 R0 (⌦, (F⇧ ), P) be such that
sup sup |Usn |  Ht
n 1 0st
where H 2 R0 (⌦, (F⇧ ), P) is an increasing process. Let X n , X be stochastic integrators such
R
R
em
em
that X n ! X. Let Z n = (U n ) dX n and W n = (U n ) dX. Then (Z n W n ) ! 0.
Proof. Note that in view of Theorem 4.29 for bounded predictable f
Z t
Z t
Z
Z
n
n
n
n
f dZ
f dW = (U ) f dX
(U n ) f dX.
0
0
4.9. The Emery Topology
149
So we need to show that (see (4.9.7)) for T < 1, > 0 and " > 0
Z t
Z t
sup P( sup | (U n ) f dX n
(U n ) f dX| > ) < ".
0tT
f :2S1
0
(4.9.15)
0
Recall, S1 is the class of predictable processes that are bounded by 1.
First, get < 1 such that
"
P(HT > ) 
2
and let a stop time be defined by
= inf{t > 0 : Ht
} ^ (T + 1).
or Ht
Then we have
P( < T ) = P(HT > ) 
"
2
P( sup |
0t ^T
P( sup |
0t ^T
em
Xn !
Z
t
n
(U ) f dX
0
t
n
h dX
0
n
n
Z
t
(4.9.16)
(U n ) f dX| > ) + P( < T )
Z
t
0
hn dX| >
0
"
)+ .
2
X, for the choice of 0 = and "0 = 2" invoking Corollary 4.97
n0 one has
Z t
Z t
"
n
sup P( sup | gdX
gdX| > ) 
2
0tT 0
g2K1
0
R
Finally, since
get n0 such that for n
Z
0
F
0
A
0tT
T
and for f 2 S1 writing hn = (U n ) f 1 1[0, ] we see that
Z t
Z t
n
n
P( sup | (U ) f dX
(U n ) f dX| > )
D
where K1 is the class of predictable processes bounded by 1. Since hn 2 K1 , using (4.9.16)
it follows that
Z t
Z t
n
n
P( sup | (U ) f dX
(U n ) f dX| > ) < ".
0tT
Note that the choice of
completing the proof.
0
0
and n0 did not depend upon f 2 K1 and hence (4.9.15) holds
Here is our main result that connects convergence in the Emery topology and stochastic
integration.
ucp
Theorem 4.101. Let Y n , Y 2 R0 (⌦, (F⇧ ), P) be such that Y n ! Y and let X n , X be
R
R
em
stochastic integrators such that X n ! X. Let Z n = (Y n ) dX n and Z = Y dX.
em
Then Z n ! Z.
150
Chapter 4. Stochastic Integration
Proof. We have seen in Theorem 4.99 that
Z
Z
n
(Y ) dX
Y dX
Thus suffices to prove that
Z
n
n
Z
em
! 0.
em
(Y ) dX
(Y n ) dX ! 0.
(4.9.17)
R
R
Let bn = dem ( (Y n ) dX n , (Y n ) dX). To prove that bn ! 0 suffices to prove the
following: (see proof of Theorem 4.46) For any subsequence {nk : k
1}, there exists
a further subsequence {mj : j
1} of {nk : k
1} (i.e. there exists a subsequence
{kj : j 1} such that mj = nkj ) such that
Z
Z
bmj = dem ( (Y mj ) dX mj , (Y mj ) dX) ! 0.
F
T
So now, given a subsequence {nk : k 1}, using ducp (Y nk , Y ) ! 0, let us choose mj = nkj
with kj+1 > kj and ducp (Y mj , Y )  2 j for each j
1. Then as seen earlier, this would
imply
1
X
m
[sup|Yt j Yt |] < 1, 8T < 1.
j=1 tT
1
X
A
Thus defining
Ht = sup [
0st j=1
mj
|Ys
Ys | + |Ys |]
R
we have that H is an r.c.l.l. process and |Y mj |  H. Then by Lemma 4.100, it follows that
Z
Z
em
mj
mj
(Y ) dX
(Y mj ) dX ! 0.
D
This proves (4.9.17) completing the proof of the Theorem.
Essentially the same arguments also proves the following:
ucp
Proposition 4.102. Let Y n , Y 2 R0 (⌦, (F⇧ ), P) be such that Y n ! Y and let X n , X 2 V
R
R
be such that dvar (X n , X) ! 0. Let Z n = (Y n )dX n and Z = Y dX. Then
dvar (Z n , Z) ! 0.
In particular,
dem (Z n , Z) ! 0.
We will now show that X 7! [X, X] and (X, Y ) 7! [X, Y ] are continuous mappings in
the Emery topology.
4.9. The Emery Topology
151
em
em
Theorem 4.103. Let X n , X, Y n , Y be stochastic integrators such that X n ! X, Y n !
Y . Then
dvar ([X n , Y n ], [X, Y ]) ! 0
(4.9.18)
and as a consequence
em
[X n , Y n ] ! [X, Y ].
(4.9.19)
Proof. Let U n = X n X. Then dem (U n , 0) ! 0 implies that ducp (U n , 0) ! 0 and hence as
R
ucp
ucp
noted earlier in Proposition 4.48 (U n )2 ! 0. Also, by Theorem 4.101, (U n ) dU n ! 0.
Since
Z
[U n , U n ] = (U n )2
ucp
! 0 and so
dvar ([X n
X, X n
X]) ! 0.
(4.9.20)
p
[U n , U n ]T [Z, Z]T
F
Now, (4.6.17) gives
Var[0,T ] ([U n , Z]) 
ucp
! 0 implies that for all stochastic integrators Z
A
and hence [U n , U n ]
(U n ) dU n ,
T
it follows that [U n , U n ]
2
dvar ([U n , Z], 0) ! 0.
Since U n = X n
X, one has [U n , Z] = [X n , Z]
[X, Z] and so
dvar ([X n , Z], [X, Z]) ! 0.
R
Noting that
(4.9.21)
D
[X n , X n ] = [X n
X, X n
X] + 2[X n , X]
[X, X]
we conclude using (4.9.20) and (4.9.21) that
dvar ([X n , X n ], [X, X]) ! 0.
(4.9.22)
Now the required relation (4.9.18) follows from (4.9.22) by polarization identity (4.6.6).
The following result is proven on similar lines.
em
em
Theorem 4.104. Let X n , X, Y n , Y be stochastic integrators such that X n ! X, Y n !
Y . Then
dvar (j(X n , Y n ), j(X, Y )) ! 0.
(4.9.23)
152
Chapter 4. Stochastic Integration
Proof. As seen in (4.6.11)
j(X n
X, X n
X)t  [X n
X, X n
X]t
and hence by Theorem 4.103,
dvar (j(X n
X, X n
X), 0) ! 0.
Now proceeding as in the proof of Theorem 4.103, we can first prove that for all stochastic
integrators Z,
dvar (j(X n , Z), j(X, Z)) ! 0.
Once again, noting that
j(X n , X n ) = j(X n
X, X n
X) + 2j(X n , X)
T
we conclude that
j(X, X)
dvar (j(X n , X n ), j(X, X)) ! 0.
F
The required result, namely (4.9.23) follows from this by polarization identity (4.6.12).
Next we will prove that (Xt ) 7! (f (t, Xt )) is a continuous mapping in the Emery
topology for smooth functions f . Here is a Lemma that would be needed in the proof.
em
A
Lemma 4.105. Let X n , Y n be stochastic integrators such that X n ! X, Y n
ucp
Suppose Z n is a sequence of r.c.l.l. processes such that Z n ! Z. Let
X
Ant =
Zsn ( X n )s ( Y n )s
em
! Y.
R
0<st
At =
X
Zs ( X)s ( Y )s .
0<st
D
Then dvar (An , A) ! 0.
Proof. Note that writing B n = j(X n , Y n ) and B = j(X, Y ), we have
Z
Z
n
n
n
A = Z dB , A = ZdB.
We have seen in (4.9.23) that dvar (B n , B) ! 0. The conclusion now follows from Proposition 4.102.
Theorem 4.106. Let X (n) = (X 1,n , X 2,n , . . . , X d,n ) be a sequence of Rd -valued r.c.l.l.
processes such that X j,n are stochastic integrators for 1  j  d, n 1. Let
em
X j,n ! X j , 1  j  d.
4.9. The Emery Topology
153
Let f 2 C 1,2 ([0, 1) ⇥ Rd ). Let
(n)
Ztn = f (t, Xt ), Zt = f (t, Xt )
em
where X = (X 1 , X 2 , . . . , X d ), Then Z n ! Z.
Proof. By Ito’s formula, (4.87), we can write
Ztn = Z0n + Ant + Ytn + Btn + Vtn , Zt = Z0 + At + Yt + Bt + Vt
=
At =
Ytn =
Z
Z
t
0
f0 (s, Xs(n) )ds
t
f0 (s, Xs )ds
0
d Z t
X
0+
j=1
Yt =
Btn
d Z
X
t
j=1
0+
d
d
(n)
fj (s, Xs )dXsj,n
T
Ant
fj (s, Xs )dXsj
1X X
=
2
t
0+
(n)
fjk (s, Xs )d[X j,n , X k,n ]s
A
j=1 k=1
Z
F
where
R
d
d Z
1X X t
Bt =
fjk (s, Xs )d[X j , X k ]s
2
0
j=1 k=1
X
(n)
n
V =
h(s, Xs(n) , Xs )
0<st
D
V =
X
h(s, Xs , Xs ).
0<st
By continuity of f0 , fj , fjk (the partial derivatives of f (t, x) w.r.t. t, xj and xj , xk respectively), and Proposition 4.48 we have
(n)
ucp
(n)
ucp
(n)
ucp
f0 (·, X· )
fj (·, X· )
em
fjk (·, X· )
!f0 (·, X· );
!fj (·, X· ), 1  j  d;
!fjk (·, X· ), 1  j, k  d.
em
Also, X (n) ! X implies that for all j, k, 1  j, k  d, [X j,n , X k,n ] ! [X j , X k ] as seen
in (4.9.19), Theorem 4.103. Thus, by Theorem 4.101, it follows that
em
em
em
An ! A, Y n ! Y, B n ! B.
(4.9.24)
154
Chapter 4. Stochastic Integration
If X n , X were continuous processes so that V n , V are identically equal to zero and the
result follows from (4.9.24). For the r.c.l.l. case, recall that h can be expressed as
d
X
h(t, y, x) =
gjk (t, y, x)(y j
xj )(y k
xk )
j,k=1
where gjk are defined by (4.8.24). Thus,
Vn =
d
X
X
(n)
gjk (s, Xs(n) , Xs )( X j,n )s ( X k,n )s ,
j,k=1 0<st
V =
d
X
X
gjk (s, Xs , Xs )( X j )s ( X k )s .
j,k=1 0<st
em
T
Now Lemma 4.105 implies that dvar (V n , V ) ! 0 and as a consequence, V n ! V .
em
Now combining this with (4.9.24) we finally get Z n ! Z.
F
Essentially the same proof (invoking Proposition 4.48) would yield the slightly stronger
result:
A
Theorem 4.107. Let X (n) = (X 1,n , X 2,n , . . . , X d,n ) be a sequence of Rd -valued r.c.l.l.
processes such that X j,n are stochastic integrators for 1  j  d, n 1 such that
em
X j,n ! X j , 1  j  d.
D
R
n
Let X = (X 1 , X 2 , . . . , X d ). Let f n , f 2 C 1,2 ([0, 1)⇥Rd ) be functions such that f n , f0n , fjn , fjk
converge to f, f0 , fj , fjk (respectively) uniformly on compact subsets of [0, 1) ⇥ Rd , where
f0n , f0 are derivatives in the t variable of f n = f n (t, x), f = f (t, x) respectively; fjn , fj
n ,f
n
n
are derivatives in xj and fjk
jk are derivatives in xj , xk of f = f (t, x), f = f (t, x)
respectively with x = (x1 , . . . xd ). Let
(n)
Ztn = f n (t, Xt ), Zt = f (t, Xt ).
em
Then Z n ! Z.
4.10
Extension Theorem
We have defined stochastic integrator as an r.c.l.l. process X for which JX defined by
e P) satisfying (4.2.3). Indeed, just
(4.2.1), (4.2.2) for f 2 S admits an extension to B(⌦,
assuming that JX satisfies
bp
f n ! 0 implies JX (f n )
ucp
!0
(4.10.1)
4.10. Extension Theorem
155
it can be shown that JX admits a required extension. The next exercise gives steps to
construct such an extension. The steps are like the usual proof of Caratheodory extension
theorem from measure theory. This exercise can be skipped on the first reading.
Exercise 4.108. Let X be an r.c.l.l. adapted process. Let JX (f ) be be defined by (4.2.1),
(4.2.2) for f 2 S. Suppose X satisfies
bp
ucp
f n 2 S, f n ! 0 implies JX (f n )
X
e P), let
and for ⇠ 2 B(⌦,
⇤
(4.10.2)
(f ) = sup{ducp (JX (g), 0) : |g|  |f |}
(4.10.3)
(⇠) = inf{
X
1
X
n
n
2 S, |⇠| 
X (g ) : g
n=1
Let
1
X
|g n |}.
e P) : 9f m 2 S s.t.
A = {⇠ 2 B(⌦,
⇤
X
n=1
f m ) ! 0}
(⇠
F
bp
(i) Let f n , f 2 S be such that f n ! f . Show that
bp
f n ! f implies JX (f n )
A
(4.10.4)
T
For f 2 S let
! 0.
ucp
! JX (f ).
(4.10.5)
(ii) Let f, f 1 , f 2 2 S and c 2 R. Show that
R
X
X
1
(f ) =
D
⇤
X
X
2
(f 1 + f 2 ) 
e P)
(iii) Show that for ⇠1 , ⇠2 2 B(⌦,
(|f |).
(cf ) = |c|
|f |  |f | )
X
X
(⇠1 + ⇠2 ) 
(|f |).
(4.10.7)
1
X
X
(4.10.6)
(f ) 
(f 1 ) +
⇤
X
(⇠1 ) +
2
X
(f ).
X
(f 2 ).
⇤
(⇠2 ).
X
(4.10.8)
(4.10.9)
(iv) Show that A is a vector space.
(v) Let f n 2 S for n
bp
1 be such that f n ! 0. Show that
X
(vi) Let hn , h 2 S for n
(f n ) ! 0.
(4.10.10)
bp
1 be such that and hn ! h Show that
X
(hn ) !
X
(h).
(4.10.11)
156
Chapter 4. Stochastic Integration
(vii) Let f n 2 S be such that f n  f n+1  K for all n, where K is a constant. Show that
8✏ > 0 9n⇤ such that for m, n n⇤ , we have
X
(f m
f n ) < ✏.
(4.10.12)
[Hint: Prove by contradiction. If not true, get ✏ > and subsequences {nk }, {mk } both
increasing to 1 such that
X
Observe that g k defined by g k = f n
e P)
(viii) For ⇠ 1 , ⇠ 2 2 B(⌦,
k
fm )
k
f m satisfy g k ! 0.]
(f n
k
✏ 8k
1.
bp
k
⇤
|⇠ 1 |  |⇠ 2 | )
X
⇤
(⇠ 1 ) 
X
(⇠ 2 ).
(
X
1
X
j=1
(x) Let g 2 S and f n 2 S, n
⇠j ) 
1
X
1 satisfy
1
X
n=1
R
X (|g|) 
[Hint: Let g n = min(|g|,
k=1 |f
k |)
⇤
X
(xii) For f 2 S show that
|f n |.
1
X
X
k=1
(4.10.15)
(|f k |).
(4.10.16)
bp
(f ) =
X
⇤(JX (f )) 
(f ).
⇤
X
(4.10.17)
(f ).
(4.10.18)
e P) is such that |⇠|  K, then for each m
(xiii) Let hm = 1[0,m] . If ⇠ 2 B(⌦,
⇤
X
(4.10.14)
and note that g n ! |g|.]
D
(xi) For f 2 S show that
Pn
(⇠ j ).
j=1
A
|g| 
Show that
⇤
X
F
⇤
T
e P) be such that ⇠ = P1 |⇠ j |. Then
(ix) Let ⇠ j , ⇠ 2 B(⌦,
j=1
(4.10.13)
(⇠) 
X
(K|hm |) + 2
m+1
.
e P) and let aj 2 R be such that aj ! 0. Show that
(xiv) Let ⇠ 2 B(⌦,
lim
j!1
⇤
X
(aj ⇠) = 0.
1 show that
(4.10.19)
(4.10.20)
4.10. Extension Theorem
157
e P) for n
(xv) Let ⇠, ⇠ n 2 B(⌦,
1 be such that ⇠ n ! ⇠ uniformly. Show that
⇤
lim
X
n!1
(⇠ n
⇠) = 0.
e P) and ⇠ n 2 A be such that limm!1
(xvi) Let ⇠ 2 B(⌦,
(xvii) Let ⇠ n 2 A be such that ⇠ n  ⇠ n+1 8n
(4.10.21)
⇤ (⇠ m
⇠) = 0. Show that ⇠ 2 A.
X
1 be such that
e P).
⇠ = lim ⇠ n 2 B(⌦,
(4.10.22)
n!1
(a) Given ✏0 > 0, show that 9g m 2 S such that g n  g n+1 8n
⇤
X
(⇠ m
n⇤ we have
1 show that 9nk such that nk > nk
(⇠ n
(d) Show that
⇤
(e) Show that ⇠ 2 A.
(f) Show that
⇤ (⇠
X
(⇠
1
X
nk
⇠ )
⇠n
k 1
(here n0 = 1) and
k
)<2
⇤
X
(⇠ n
m+1
m=k
.
m
⇠n )  2
k+1
.
(4.10.23)
A
X
k
1
F
⇤
X
⇠ n )  ✏0 .
⇠ n ) < ✏.
[Hint : Use (a) above along with (vii).]
(c) For k
⇤ (g n
X
T
(b) Given ✏ show that 9n⇤ such that for n, m
1 and
⇠ n ) ! 0.
R
[Hint: Use (d) above to conclude that every subsequence of an =
further subsequence converging to 0.]
⇤ (⇠
X
⇠ n ) has a
D
e P).
(xviii) Show that A = B(⌦,
[Hint: Use Monotone Class Theorem 2.61].
e P) be a sequence such that ⌘ n
(xix) Let ⌘ n 2 B(⌦,
⌘ n+1
Show that
⇤ n
(⌘ ) ! 0.
[Hint: Let ⇠ n = ⌘ 1
0 for all n with limn ⌘ n = 0.
⌘ n and use (f ) above].
bp
e P) be such that f n ! f . Show that
(xx) Let f n , f 2 B(⌦,
⇤
[Hint: Let g n = supm
n |f
m
(f n
f ) ! 0.
f |. Then use g n # 0 and |f m
(4.10.24)
f |  g m .]
158
Chapter 4. Stochastic Integration
e P) be such that f n bp
(xxi) Let f n , f 2 B(⌦,
! f . Show that JX (f n ) is Cauchy in ducp -metric.
D
R
A
F
T
(xxii) Show that X is a stochastic integrator.
Chapter 5
Semimartingales
D
R
A
F
T
The reader would have noticed that in the development of stochastic integration in the
previous chapter, we have not talked about either martingales or semi martingales.
A semimartingale is any process which can be written as a sum of a local martingale
and a process with finite variation paths.
The main theme of this chapter is to show that the class of stochastic integrators is the
same as the class of semimartingales, thereby showing that stochastic integral is defined
for all semimartingales and the Ito formula holds for them. This is the Dellacherie-MeyerMokobodzky-Bichteler Theorem.
Traditionally, the starting point for integration with respect to square integrable martingales is the Doob-Meyer decomposition theorem. We follow a di↵erent path, proving
that for a square integrable martingale M , the quadratic variation [M, M ] can be defined
directly and then Xt = Mt2 [M, M ]t is itself a martingale. This along with Doob’s maximal inequality would show that square integrable martingales (and locally square integrable
martingales) are integrators. We would then go on to show that a local martingale (and
hence any semimartingale) is an integrator.
Next we show that every stochastic integrator is a semimartingale, thus proving a weak
version of the Dellacherie-Meyer-Mokobodzky- Bichteler Theorem. Subsequently, we prove
the full version of this result.
5.1
Notations and terminology
We begin with some definitions.
Definition 5.1. An r.c.l.l. adapted process M is said to be a square integrable martingale if
159
160
Chapter 5. Semimartingales
M is a martingale and
E[Mt2 ] < 1 8t < 1.
Exercise 5.2. Let M be a square integrable martingale. Show that
(i) sup0tT E[Mt2 ] < 1 for each T < 1.
(ii) E[sup0tT Mt2 ] < 1 for each T < 1.
Definition 5.3. An r.c.l.l. adapted process L is said to be a local martingale if there exist stop
n
times ⌧ n " 1 such that for each n, the process M n defined by M n = L[⌧ ] , i.e. Mtn = Lt^⌧ n
is a martingale.
n and ⌧ n = n if U  n. Let M n = L[⌧
n]
be the stopped process. Show
A
Let ⌧ n = 1 if U
that
F
T
Exercise 5.4. Let W be the Brownian motion and let U = exp(W12 ). Let
8
<W
if t < 1
t
Lt =
:W + U (W
W1 ) if t 1.
1
t
(i) For each n, ⌧ n is a stop time for the filtration (F⇧W ).
R
(ii) ⌧ n " 1.
(iii) M n is a martingale (w.r.t. the filtration (F⇧W )).
D
(iv) E[|Lt | ] = 1 for t > 1
This gives an example of a local martingale that is not a martingale.
For a local martingale L, one can choose the localizing sequence {⌧ n } such that ⌧ n  n
n
and then M n = L[⌧ ] is uniformly integrable.
Here is a simple condition under which a local martingale is a martingale.
Lemma 5.5. Let an r.c.l.l. process L be a local martingale such that
E[ sup |Ls | ] < 1 8t < 1.
0st
Then L is a martingale.
(5.1.1)
5.1. Notations and terminology
161
n
Proof. Let ⌧ n " 1 such that for each n, the process M n defined by M n = L[⌧ ] is a
martingale. Then Mtn converges to Lt pointwise and (5.1.1) implies that sup0st |Ls |
serves as a dominating function. Thus, for each t, Mtn converges to Lt in L1 (P). Now the
required result follows using Theorem 2.21.
Definition 5.6. An r.c.l.l. adapted process N is said to be a locally square integrable martingale if there exist stop times ⌧ n " 1 such that for each n, the process M n defined by
n
M n = N [⌧ ] , i.e. Mtn = Nt^⌧ n is a square integrable martingale.
T
Exercise 5.7. Show that the process L constructed in Exercise 5.4 is a locally square integrable martingale.
F
Definition 5.8. An r.c.l.l. adapted process X is said to be a semimartingale if X can be
written as X = M + A where M is an r.c.l.l. local martingale and A is an r.c.l.l. process whose
paths have finite variation on [0, T ] for all T < 1, i.e. Var[0,T ] (A) < 1 for all T < 1.
R
A
Let us denote by M the class of all r.c.l.l. martingales M with M0 = 0, M2 the class
of all r.c.l.l. square integrable martingales with M0 = 0. We will also denote by Mloc the
class of r.c.l.l. local martingales with M0 = 0 and M2loc the class of r.c.l.l. locally square
integrable martingales with M0 = 0. Thus, M 2 M2loc (Mloc ) if there exist stop times ⌧ n
n
increasing to infinity such that the stopped processes M [⌧ ] belong to M2 (respectively
belong to M).
D
Let Mc be the class of all continuous martingales M with M0 = 0, Mc,loc be the class of
all continuous local martingales M with M0 = 0 and M2c be the class of square integrable
continuous martingales M with M0 = 0.
Exercise 5.9. Show that Mc,loc ✓ M2loc .
Thus X is a semimartingale if we can write X = M + A where M 2 Mloc and A 2
V. We will first show that semimartingales are stochastic integrators. Recall that all
semimartingales and stochastic integrators are by definition r.c.l.l. processes. We begin
by showing that square integrable r.c.l.l. martingales are stochastic integrators. Usually
this step is done involving Doob-Meyer decomposition theorem. We bypass the same by a
study of quadratic variation as a functional on the path space
162
5.2
Chapter 5. Semimartingales
The Quadratic Variation Map
Let D([0, 1), R) denote the space of r.c.l.l. functions on [0, 1). Recall our convention that
for ↵ 2 D([0, 1), R), ↵(t ) denotes the left limit at t (for t > 0) and ↵(0 ) = 0 and
↵(t) = ↵(t) ↵(t ). Note that by definition, ↵(0) = ↵(0).
Exercise 5.10. Suppose fn 2 D([0, 1), R) are such that fn converges to a function f
uniformly on compact subsets of [0, 1), i.e. sup0tT |fn (t) f (t)| converges to zero for all
T < 1. Show that
(i) f 2 D([0, 1), R).
(iii) fn (s ) converges to f (s ) for all s 2 [0, 1).
(iv) ( fn )(s) converges to ( f )(s) for all s 2 [0, 1).
T
(ii) If sn 2 [0, s) converges to s 2 [0, 1) then fn (sn ) converges to f (s ).
A
F
We will now define quadratic variation (↵) of a function ↵ 2 D([0, 1), R).
For each n 1; let {tni (↵) : i 1} be defined inductively as follows : tn0 (↵) = 0 and
having defined tni (↵), let
tni+1 (↵) = inf{t > tni (↵) : |↵(t)
↵(tni (↵))|
2
n
↵(tni (↵))|
or |↵(t )
2
n
}.
R
If limi tni (↵) = t⇤ < 1, then the function ↵ cannot have a left limit at t⇤ . Hence for each
↵ 2 D([0, 1), R) , tni (↵) " 1 as i " 1 for each n. Let
n (↵)(t)
=
1
X
i=0
(↵(tni+1 (↵) ^ t)
↵(tni (↵) ^ t))2 .
(5.2.1)
D
Since tni (↵) increases to infinity, for each ↵ and n fixed, the infinite sum appearing above
is essentially a finite sum and hence n (↵) is itself an r.c.l.l. function. The space D =
D([0, 1), R) is equipped with the topology of uniform convergence on compact subsets
e denote the set of ↵ 2 D such that n (↵) converges in the ucc
(abbreviated as ucc). Let D
topology and
8
<lim
e
if ↵ 2 D
n n (↵)
(↵) =
(5.2.2)
:0
e
if ↵ 62 D.
Here are some basic properties of the quadratic variation map
e
Lemma 5.11. For ↵ 2 D
.
5.2. The Quadratic Variation Map
(i)
(↵) is an increasing r.c.l.l. function.
(↵)(t) = ( ↵(t))2 for all t 2 (0, 1).
(ii)
(iii)
163
P
st (
(iv) Let
↵(s))2 < 1 for all t 2 (0, 1).
c (↵)(t)
=
P
(↵)(t)
↵(s))2 . Then
0<st (
c (↵)
is a continuous function.
Proof. For (i), note that for s  t, if tnj  s < tnj+1 , then |(↵(s)
n (↵)(t)
=
j 1
X
i=0
jX
1
(↵(tni+1 (↵))
↵(tni (↵)))2 + (↵(s)
(↵(tni+1 (↵))
↵(tni (↵)))2
i=0
+
1
X
i=j
(↵(tni+1 (↵) ^ t)
n (↵)(s)

n (↵)(t)
and
↵(tnj ))2
↵(tni (↵) ^ t))2
F
and hence
n,
T
n (↵)(s) =
↵(tnj ))|  2
+2
2n
.
(5.2.3)
D
R
A
Compare with (4.6.4). Thus (5.2.3) is valid for all n 1 and s  t and hence it follows that
the limiting function (↵) is an increasing function. Uniform convergence of the r.c.l.l.
function n (↵) to (↵) implies that (↵) is an r.cl.l. function.
For (ii), it is easy to see that the set of points of discontinuity of n (↵) are contained
in the set of points of discontinuity of ↵ for each n. Uniform convergence of n (↵)(t) to
(↵)(t) for t 2 [0, T ] for every T < 1 implies that the same is true for (↵) i.e. for t > 0,
(↵)(t) 6= 0 implies that ↵(t) 6= 0.
On the other hand, let t > 0 be a discontinuity point for ↵. Let us note that by the
definition of tnj (↵),
|↵(u)
↵(v)|  2.2
n
8u, v 2 [tnj (↵), tnj+1 (↵)).
(5.2.4)
Thus for n such that 2.2 n < (↵)(t), t must be equal to tnk (↵) for some k 1 since(5.2.4)
implies ↵(v) 2.2 n for any v 2 [j (tnj (↵), tnj+1 (↵)). Let sn = tnk 1 (↵) where k is such
that t = tnk (↵). Note that sn < t. Let s⇤ = lim inf n sn . We will prove that
lim ↵(sn ) = ↵(t ), lim
n
n
n (↵)(sn )
=
(↵)(t ).
(5.2.5)
If s⇤ = t, then sn < t for all n
1 implies sn ! t and (5.2.5) follows from uniform
convergence of n (↵) to (↵) on [0, t] (see Exercise 5.10).
164
Chapter 5. Semimartingales
If s⇤ < t, using (5.2.4) it follows that |↵(u) ↵(v)| = 0 for u, v 2 (s⇤ , t) and hence
the function ↵(u) is constant on the interval (s⇤ , t) and implying that sn ! s⇤ . Also,
↵(s⇤ ) = ↵(t ) and (↵)(s⇤ ) = (↵)(t ). So if ↵ is continuous at s⇤ , once again uniform
convergence of n (↵) to (↵) on [0, t] shows that (5.2.5) is valid in this case too.
Remains to consider the case s⇤ < t and ↵(s⇤ ) = > 0. In this case, for n such that
2.2 n < , sn = s⇤ and uniform convergence of n (↵) to (↵) on [0, t] shows that (5.2.5)
is true in this case as well.
We have (for large n)
n (↵)(t)
=
n (↵)(sn )
+ (↵(sn )
↵(t))2
(5.2.6)
and hence (5.2.5) yields
(↵)(t ) + [ ↵(t)]2
(↵)(t) =
0<st
A
The last part, (iv) follows from (ii), (iii).
F
T
completing the proof of (ii).
(iii) follows from (i) and (ii) since for an increasing function that is non-negative at
zero, the sum of jumps up to t is at most equal to its value at t:
X
( ↵(s))2  (↵)(t).
R
Remark 5.12.
is the quadratic variation map. It may depend upon the choice of the
partitions. If instead of 2 n , we had used any other sequence {"n }, it would yield another
mapping ˜ which will have similar properties. Our proof in the next section will show that if
P
n "n < 1, then for a square integrable local martingale (Mt ),
(M.(!)) = ˜ (M.(!)) a.s. P.
D
We note two more properties of the quadratic variation map
variation Var(0,T ] (↵) of ↵ on the interval (0, T ] is defined by
m
X1
Var(0,T ] (↵) = sup{
j=0
|↵(sj+1 )
. Recall that the total
↵(sj )| : 0  s1  s2  . . . sm = T, m
1}.
If Var(0,T ] (↵) < 1, ↵ is said to have finite variation on [0, T ] and then on [0, T ] it can be
written as di↵erence of two increasing functions.
Lemma 5.13. The quadratic variation map
satisfies the following.
e and 0 < s < 1 fixed, let ↵s 2 D be defined by: ↵s (t) = ↵(t ^ s). Then
(i) For ↵ 2 D
e
↵s 2 D.
5.3. Quadratic Variation of a Square Integrable Martingale
165
e for all k, then
(ii) For ↵ 2 D and sk " 1, ↵k be defined via ↵k (t) = ↵(t ^ sk ). If ↵k 2 D
e and
↵2D
(↵)(t ^ sk ) = (↵k )(t) , 8t < 1, 8k 1.
(5.2.7)
(iii) Suppose ↵ is continuous, and Var(0,T ] (↵) < 1. Then
(↵)(t) = 0, 8t 2 [0, T ].
Proof. (i) is immediate. For (ii), it can be checked from the definition that
^ sk ) =
n (↵)(t
n (↵
k
)(t) , 8t.
(5.2.8)
e it follows that n (↵)(t) converges uniformly on [0, sk ] for every k and hence
Since ↵k 2 D,
e and that (5.2.7) holds.
using (5.2.8) we conclude that ↵ 2 D
For (iii), note that ↵ being a continuous function,
↵(tni (↵) ^ t)|  2
for all i, n and hence we have
1
X
i=0
2
n
⇥
1
X
i=0
|↵(tni+1 (↵) ^ t)
↵(tni (↵) ^ t)|
⇥ Var[0,T ] (↵).
(↵)(t) = 0 for t 2 [0, T ].
Quadratic Variation of a Square Integrable Martingale
R
5.3
n
↵(tni (↵) ^ t))2
A
This shows that
2
(↵(tni+1 (↵) ^ t)
F
n (↵)(t) =
n
T
|↵(tni+1 (↵) ^ t)
The next lemma connects the quadratic variation map
and r.c.l.l. martingales.
D
Lemma 5.14. Let (Nt , Ft ) be an r.c.l.l. martingale such that E(Nt2 ) < 1 for all t > 0.
Suppose there is a constant C < 1 such that with
⌧ = inf{t > 0 : |Nt |
C or |Nt |
C}
one has
Nt = Nt^⌧ .
Let
At (!) = [ (N.(!))](t).
Then (At ) is an (Ft ) adapted r.c.l.l. increasing process such that Xt := Nt2
martingale.
At is also a
166
Chapter 5. Semimartingales
Proof. Let
n (↵)
and tni (↵) be as in the previous section.
Ant (!) =
n
i (!)
=
n (N.(!))(t)
tni (N.(!))
Ytn (!) = Nt2 (!)
It is easy to see that for each n, {
n
i
Ant =
:i
1
X
N02 (!)
Ant (!)
1} are stop times (see theorem 2.44) and that
(N
i=0
n
i (!)
(5.3.1)
N
n
i+1 ^t
n
i ^t
)2 .
T
Further, for each n,
increases to 1 as i " 1.
We will first prove that for each n, (Ytn ) is an (Ft )-martingale. Using the identity
b2 a2 (b a)2 = 2a(b a), we can write
1
X
n ^t
Ytn = Nt2 N02
(N i+1
N in ^t )2
i=0
(N
i=0
1
X
=2
i=0
N
n
i+1 ^t
N
n
i ^t
(N
2
n
i ^t
Xtn,i = N
Then
)
(N
i=0
N
n
i+1 ^t
n
i ^t
(N
R
Ytn = 2
n
i+1 ^t
D
|N
C (x)
N
n
i ^t
N
n
i+1 ^t
n
i ^t
)2
n
i ^t
)
n
i+1 ^t
1
X
N
n
i ^t
).
Xtn,i .
(5.3.2)
i=0
Noting that for s < ⌧ , |Ns |  C and for s
Thus, writing
1
X
A
Let us define
2
F
=
1
X
⌧ , Ns = N , it follows that
| > 0 implies that |N
n
i ^t
|  C.
= max{min{x, C}, C} (x truncated at C), we have
Xtn,i =
C (N
n
i ^t
)(N
n
i+1 ^t
N
n
i ^t
)
and hence, Xtn,i is a martingale. Using the fact that Xtn,i is Ft^
E(Xtn,i |Ft^ in ) = 0, it follows that for i 6= j,
E[Xtn,i Xtn,j ] = 0.
(5.3.3)
n
i+1
measurable and that
(5.3.4)
Also, using (5.3.3) and the fact that N is a martingale we have
E(Xtn,i )2 C 2 E(N
n
i+1 ^t
n ^t
=C 2 E(N 2i+1
N . in ^t )2
N 2in ^t ).
(5.3.5)
5.3. Quadratic Variation of a Square Integrable Martingale
167
Using (5.3.4) and (5.3.5), it follows that for s  r,
E(
r
X
i=s
N 2sn ^t ).
(5.3.6)
increases to 1 as i tends to infinity, E(N 2sn ^t ) and E(N 2n ^t ) both tend to E[Nt2 ]
r+1
P
as r, s tend to 1 and hence ri=1 Xtn,i converges in L2 (P). In view of (5.3.2), one has
2
r
X
i=1
and hence
For n
(Ytn )
Xtn,i ! Ytn in L2 (P) as r ! 1
is an (Ft ) - martingale for each n
1, define a process
Nn
by
Ntn = N
n
i
Observe that by the choice of {
1.
:i
|Nt
n
i
if
n
i
1}, one has
Ntn |  2
n
i+1 .
t<
T
Since
n
i
n ^t
Xtn,i )2  C 2 E(N 2r+1
n
for all t.
(5.3.7)
H(!) = {
n
i (!)
F
For now let us fix n. For each ! 2 ⌦, let us define
1} [ {
: i
n+1
(!)
i
: i
1}
(5.3.8)
A
It may be noted that for ! such that t 7! Nt (!) is continuous, each jn (!) is necessarily
equal to in+1 (!) for some i, but this need not be the case when t 7! Nt (!) has jumps. Let
⇣0 (!) = 0 and for j 0, let
It can be verified that
R
⇣j+1 (!) = inf{s > ⇣j (!) : s 2 H(!)}.
D
{⇣i (!) : i
1} = {
n
i (!)
Since
n
k,
n+1
j
1} [ {
n+1
(!)
i
: i
1}.
1, t < 1. Let
To see that each ⇣i is a stop time, fix i
Akj = {(
: i
n
k
^ t) 6= (
n+1
j
^ t)}.
are stop times, Akj 2 Ft for all k, j. It is not difficult to see that
{⇣i  t} = [ik=0 ({
n
i k
 t} \ Bk )
where B0 = ⌦ and for 1  k  i,
Bk = [0<j1 <j2 <...<jk ( (\il=0k \km=1 Aljm ) \ {
and hence ⇣i is a stop time.
n+1
jk
 t})
(5.3.9)
168
Chapter 5. Semimartingales
Using (5.3.9) and using the fact that Ntn = Nt^ jn for jn  t <
and Y n+1 as
1
X
n
n
Yt =
2Nt^⇣
(Nt^⇣j+1 Nt^⇣j ),
j
Ytn+1 =
j=0
1
X
n+1
2Nt^⇣
(Nt^⇣j+1
j
n
j+1 ,
one can write Y n
Nt^⇣j ).
j=0
Hence
Ytn+1
Ytn = 2
1
X
Ztn,j
(5.3.10)
j=0
where
n
Nt^⇣
)(Nt^⇣j+1
j
Also, using (5.3.7) one has
|Ntn+1
Ntn |  |Ntn+1
Nt | + |Nt
Ntn |  2
Nt^⇣j ).
T
n+1
Ztn,j = (Nt^⇣
j
(n+1)
+2
n
 2.2
n
(5.3.11)
R
and hence (using (5.3.12))
A
F
and hence (using that (Ns ) is a martingale), one has
4
4
E[(Ztn,j )2 ]  2n E[(Nt^⇣j+1 Nt^⇣j )2 ] = 2n E[(Nt^⇣j+1 )2 (Nt^⇣j )2 ].
(5.3.12)
2
2
n
Easy to see that E(Ztn,j |Ft^ jn ) = 0 and Ztn,j is Ft^ j+1
measurable. It then follows that
for i 6= j
E[Ztn,j Ztni ] = 0
D
E(Ytn+1
Ytn )2
= 4E[(
= 4E[

1
X
Ztn,j )2 ]
j=0
1
X
(Ztn,j )2 ]
j=0
1
X
16
22n
(5.3.13)
E[(Nt^⇣j+1 )2
(Nt^⇣j )2 ]
j=0
16
E[(Nt )2 ].
22n
Thus, recalling that Ytn+1 , Ytn are martingales, it follows that Ytn+1 Ytn is also a martingale
and thus invoking Doob’s maximal inequality, one has (using (5.3.13))

E[supsT |Ysn+1
Ysn |2 ]  4E(YTn+1 YTn )2
64
 2n E[NT2 ].
2
(5.3.14)
5.3. Quadratic Variation of a Square Integrable Martingale
Thus, for each n
1,
k [supsT |Ysn+1
It follows that
⇠=
1
X
Ysn |] k2 
sup|Ysn+1
n=1sT
Hence (Ysn )
169
8
kNT k2 .
2n
(5.3.15)
Ysn | < 1 a.s.
as k⇠k2 < 1 by (5.3.15).
converges uniformly in s 2 [0, T ] for every T a.s.
to an r.c.l.l. process say (Ys ). As a result, (Ans ) also converges uniformly in s 2 [0, T ] for
every T < 1 a.s. to say (Ãs ) with Yt = Nt2 N02 Ãt . Further, (5.16) also implies that
for each s, convergence of Ysn to Ys is also in L2 and thus (Yt ) is a martingale. Since Ans
converges uniformly in s 2 [0, T ] for all T < 1 a.s., it follows that
e =1
P(! : N· (!) 2 D)
N02
At is a martingale. This
T
and Ãt = At . We have already proven that Yt = Nt2
completes the proof.
F
Exercise 5.15. Use completeness of the underlying -field to show that the set {! : N· (!) 2
e appearing above is measurable.
D}
R
A
We are now in a position to prove an analogue of the Doob-Meyer decomposition
theorem for the square of an r.c.l.l. locally square integrable martingale. We will use the
notation [N, N ] for the process A = (N.(!)) of the previous result and call it quadratic
variation of N . We will later show that square integrable martingales and locally square
integrable martingales are stochastic integrators. Then it would follow that the quadratic
variation defined for a stochastic integrator X via (4.6.2) agrees with the definition given
below for a square integrable martingale and a locally square integrable martingale.
D
Theorem 5.16. Let (Mt , Ft ) be an r.c.l.l. locally square integrable martingale i.e. there
exist stop times ⇣n increasing to 1 such that for each n, Mtn = Mt^⇣n is a martingale with
E[(Mt^⇣n )2 ] < 1 for all t, n. Let
[M, M ]t (!) = [ (M.(!))](t).
(5.3.16)
Then
(i) ([M, M ]t ) is an (Ft ) adapted r.c.l.l. increasing process such that Xt = Mt2
is also a local martingale.
(ii)
P( [M, M ]t = ( Mt )2 , 8t > 0) = 1.
[M, M ]t
170
Chapter 5. Semimartingales
(iii) If (Bt ) is an r.c.l.l. adapted increasing process such that B0 = 0 and
P( Bt = ( Mt )2 , 8t > 0) = 1
and Vt = Mt2
Bt is a local martingale, then P(Bt = [M, M ]t , 8t) = 1.
(iv) If M is a martingale and E(Mt2 ) < 1 for all t, then E([M, M ]t ) < 1 for all t and
Xt = Mt2 [M, M ]t is a martingale.
(v) If E([M, M ]t ) < 1 for all t and M0 = 0, then E(Mt2 ) < 1 for all t, (Mt ) is a
martingale and Xt = Mt2 [M, M ]t is a martingale.
1, let ⌧k be the stop time defined by
⌧k = inf{t > 0 : |Mt |
k or |Mt |
k} ^ ⇣k ^ k.
T
Proof. For k
F
Then ⌧k increases to 1 and Mtk = Mt^⌧k is a martingale satisfying conditions of Lemma
5.14 with C = k and ⌧ = ⌧k . Hence Xtk = (Mtk )2 [M k , M k ]t is a martingale, where
[M k , M k ]t = (M·k (!))t . Also,
e = 1, 8k
P({! : M·k (!) 2 D})
1.
(5.3.17)
A
Since Mtk = Mt^⌧k it follows from Lemma 5.13 that
e =1
P({! : M· (!) 2 D})
and
(5.3.18)
R
P({! : [M k , M k ]t (!) = [M, M ]t^⌧k (!) (!) 8t}) = 1.
D
It follows that Xt^⌧k = Xtk a.s. and since X k is a martingale for all k, it follows that
Xt is a local martingale. This completes the proof of part (i).
Part (ii) follows from Lemma 5.11.
For (iii), note that from part (ii) and the hypothesis on B it follows that
Ut = [M, M ]t
Bt
is a continuous process. Recalling Xt = Mt2 [M, M ]t and Vt = Mt2 Bt are local
martingales, it follows that Ut = Vt Xt is also a local martingale with U0 = 0. By part (i)
above, Wt = Ut2 [U, U ]t is a local martingale. On the other hand, Ut being a di↵erence
of two increasing functions has bounded variation, Var(0,T ] (U ) < 1. Continuity of U and
part (iii) of Lemma 5.13 gives
[U, U ]t = 0 8t.
5.3. Quadratic Variation of a Square Integrable Martingale
Hence Wt = Ut2 is a local martingale. Now if
Wt^ k is a martingale for k 1, then we have
k
171
are stop times increasing to 1 such that
2
E[Wt^ k ] = E[Ut^
] = E[U02 ] = 0.
k
2
and hence Ut^
= 0 for each k. This yields Ut = 0 a.s. for every t. This completes the
k
proof of (iii).
For (iv), we have proven in (i) that Xt = Mt2 [M, M ]t is a local martingale. Let k
be stop times increasing to 1 such that Xtk = Xt^ k are martingales. Hence, E[Xtk ] = 0,
or
2
E([M, M ]t^ k ) = E(Mt^
)
k
E(M02 ).
(5.3.19)
2
E([M, M ]t^ k )  E(Mt^
)  E(Mt2 ).
k
(5.3.20)
T
Hence
Now Fatou’s lemma (or Monotone convergence theorem) gives
(5.3.21)
F
E([M, M ]t )  E(Mt2 ) < 1.
A
2
It also follows that Mt^
converges to Mt2 in L1 (P) and [M, M ]t^ k converges to [M, M ]t
k
in L1 (P). Thus It follows that Xtk converges to Xt in L1 (P) and hence (Xt ) is a martingale.
For (v) let k be as in part (iv). Using M0 = 0, that [M, M ]t is increasing and (5.3.19)
we conclude
R
2
] = E([M, M ]t^ k )
E[Mt^
k
 E([M, M ]t )
D
Now using Fatou’s lemma, one gets
E[Mt2 ]  E([M, M ]t ) < 1.
Now we can invoke part (iv) to complete the proof.
Corollary 5.17. For an r.c.l.l. martingale M with M0 = 0 and E[MT ]2 < 1, one has
E[ [M, M ]T ]  E[ sup |Ms |2 ]  4E[ [M, M ]T ]
(5.3.22)
0sT
Proof. Let Xt = Mt2 [M, M ]t . As noted above X is a martingale. Since X0 = 0, it
follows that E[XT ] = 0 and thus
E[ [M, M ]T ] = E[MT2 ].
(5.3.23)
The inequality (5.3.22) now follows from Doob’s maximal inequality, Theorem 2.24.
172
Chapter 5. Semimartingales
Corollary 5.18. For an r.c.l.l. locally square integrable martingale M , for any stop time
, one has
E[ sup |Ms |2 ]  4E[ [M, M ] ]
(5.3.24)
0s
Proof. If ⌧n " 1, then using (5.3.22) for the square integrable martingale Mtn = Mt^⌧n we
get
E[
sup
0s ^⌧n
|Ms |2 ]  4E[ [M, M ]
^⌧n ]
 4E[ [M, M ] ].
Now the required result follows by Fatou’s lemma.
T
Theorem 5.19. Let M be a continuous local martingale with M0 = 0. If M 2 V then
Mt = 0 for all t.
F
Proof. Invoking (iii) in Lemma 5.13, we conclude that [M, M ]t = 0 for all t and thus the
conclusion follows from Corollary 5.18.
R
A
Remark 5.20. The pathwise formula for quadratic variation of a continuous local martingale
M was proven in Karandikar [33], but the proof required the theory of stochastic integration.
A proof involving only Doob’s inequality as presented above for the case of continuous local
martingales was the main theme of Karandikar [34]. The formula for r.c.l.l. case was given in
Karandikar [37] but the proof required again the theory of stochastic integration. The treatment
given above is adapted from Karandikar - Rao [41].
D
Exercise 5.21. If P is Weiner measure on C([0, 1), Rd ) and Q is a probability measure
absolutely continuous w.r.t. P such that the coordinate process is a local martingale (in the
sense that each component is a local martingale), then P = Q.
Hint: Use Levy’s characterization of Brownian motion.
For locally square integrable r.c.l.l. martingales M, N , we define cross-quadratic variation [M, N ] by the polarization identity as in the case of stochastic integrators (see
(4.6.6))
1
[M, N ]t = ([M + N, M + N ]t
[M N, M N ]t )
(5.3.25)
4
and then it can be checked that [M, N ] is the only process B in V0 such that Mt Nt
is a local martingale and P( ( B)t = ( M )t ( N )t 8t) = 1.
Bt
5.4. Square Integrable Martingales are Stochastic Integrators
5.4
173
Square Integrable Martingales are Stochastic Integrators
The main aim of this section is to show that square integrable martingales are stochastic
integrators.
The treatment is essentially classical, as in Kunita-Watanabe [44], but with an exception. The role of hM, M i - the predictable quadratic variation in the Kunita-Watanabe
treatment is here played by the quadratic variation [M, M ].
Recall that M2 denotes the class of r.c.l.l. martingales M such that E[Mt2 ] < 1 for all
t < 1 with M0 = 0.
T
Lemma 5.22. Let M, N 2 M2 and f, g 2 S. Let X = JM (f ) and Y = JN (g). Let
Rt
Z t = Xt Yt
0 fs gs d[M, N ]s . Then X, Y, Z are martingales.
0tT
A
F
Proof. The proof is almost exactly the same as proof of Lemma 3.8 and it uses Mt Nt
[M, N ]t is a martingale along with Theorem 2.57, Corollary 2.58 and Theorem 2.59.
Rt
Corollary 5.23. Let M 2 M2 and f 2 S. Then Yt = 0 f dM and Zt = (Yt )2
Rt 2
0 fs d[M, M ]s are martingales and
Z t
Z T
2
E[ sup | f dM | ]  4E[
fs2 d[M, M ]s ].
(5.4.1)
0
0
R
Proof. Lemma 5.22 gives Y, Z are martingales. The estimate (5.4.1) now follows from
Doob’s inequality.
D
Theorem 5.24. Let M 2 M2 . Then M is a stochastic integrator. Further, for f 2
R
Rt 2
e P), the processes Yt = t f dM and Zt = Y 2
B(⌦,
t
0
0 fs d[M, M ]s are martingales,
Rt 2
[Y, Y ]t = 0 fs d[M, M ]s and
Z t
Z T
2
E[ sup | f dM | ]  4E[
fs2 d[M, M ]s ], 8T < 1.
(5.4.2)
0tT
0
0
Proof. Fix T < 1. Suffices to prove the result for the case when Mt = Mt^T . The rest
e = [0, 1) ⇥ ⌦ and P is the
follows by localization. See Theorem 4.45. Recall that ⌦
e Let µ be the measure on (⌦,
e P) defined for A 2 P
predictable -field on ⌦.
Z Z T
µ(A) = [
1A (!, s)d[M, M ]s (!)]dP(!).
(5.4.3)
0
Note that
e = E[ [M, M ]T ] = E[|MT |2 ] < 1
µ(⌦)
174
Chapter 5. Semimartingales
e P) the norm on L2 (⌦,
e P, µ) is given by
and for f 2 B(⌦,
s
Z T
kf k2,µ = E[
fs2 d[M, M ]s ].
(5.4.4)
0
e P) ✓ L2 (⌦,
e P, µ). Since (S) = P, it follows from Theorem 2.65 that S is
Clearly, B(⌦,
e P, µ). Thus, given f 2 B(⌦,
e P), we can get f n 2 S such that
dense in L2 (⌦,
Letting Ytn =
Rt
0
kf
f n k2,µ  2
n 1
.
(5.4.5)
f n dM for t  T and Ytn = YTn for t > T one has for m, n
E[ sup |Ytn
Ytm |2 ]  4kf m
0tT
f n k22,µ  4.4
n
k
.
T
It then follows that (as in the proof of Lemma 3.10) Ytn converges uniformly in t to Yt
(a.s.), where Y is an r.c.l.l. adapted process with Yt = Yt^T . For any g 2 S, using the
estimate (5.4.1) for f n g, we get
Z t
E[ sup |Ytn
gdM |2 ]  4kf n gk22,µ
0tT
F
0
A
and taking limit as n tends to infinity in the inequality above we get
Z t
E[ sup |Yt
gdM |2 ]  4kf gk22,µ .
0tT
(5.4.6)
0
R
Rt
Let us denote Y as JM (f ). The equation (5.4.6) implies that for f 2 S, JM (f ) = 0 f dM .
Also, (5.4.6) implies that the process Y does not depend upon the choice of the particular
e P), a sequence hm 2 S approximating
sequence {f n } in (5.4.5). Further, taking h 2 B(⌦,
h, using (5.4.6) for hm and taking limit as m ! 1
E[ sup |(JM (f ))t
D
0tT
(JM (h))t |2 ]  4kf
bp
hk22,µ .
(5.4.7)
The estimate (5.4.7) implies that if fn ! f , then JM (f n ) converges to JM (f ) in ucp
topology and thus M is a stochastic integrator.
The estimate (5.4.2) follows from (5.4.7) by taking h = 0.
Rt
RT
Remains to show that Yt = 0 f dM and Zt = Yt2 0 fs2 d[M, M ]s are martingales. We
RT n 2
have seen in Corollary 5.23 that Y n and Ztn = (Ytn )2
0 (f )s d[M, M ]s are martingales.
n
2
1
Here Yt converges to Yt in L (P) (and hence in L (P)) and so Y is a martingale. Further,
(Ytn )2 ! (Yt )2 in L1 (P) and moreover kf n f k2,µ ! 0 implies
Z t
E[ |(fsn )2 fs2 |d[M, M ]s ] ! 0
0
5.4. Square Integrable Martingales are Stochastic Integrators
and thus
E|
So
Ztn
converges to Zt in
Z
t
0
L1 (P)
(
Z
((fsn )2
175
fs2 )d[M, M ]s | ! 0.
and thus Z is also a martingale. Since
fs2 d[M, M ] ) = f 2 ( M )2 = ( Y )2
using part (iii) of Theorem 5.16, it now follows that [Y, Y ]t =
Rt
Let us now introduce a class of integrands f such that
integrable martingale.
R
0
fs2 d[M, M ]s .
f dM is a locally square
T
Definition 5.25. For M 2 M2loc let L2m (M ) denote the class of predictable processes f such
that there exist stop times k " 1 with
Z k
E[
fs2 d[M, M ]s ] < 1 for k 1.
(5.4.8)
F
0
We then have the following.
A
Theorem 5.26. Let M 2 M2loc i.e. M be a locally square integrable r.c.l.l. martingale with
M0 = 0. Then M is a stochastic integrator,
R
L2m (M ) ✓ L(M )
(5.4.9)
R
t
and for f 2 L2m (M ), the process Yt = 0 f dM is a locally square integrable martingale and
R
Rt 2
t 2
Ut = Yt2
f
d[M,
M
]
is
a
local
martingales,
[Y,
Y
]
=
s
s
t
0
0 fs d[M, M ]s . Further, for
any stop time ,
Z
Z
t
D
E[ sup |
0t
0
f dM |2 ]  4E[
0
fs2 d[M, M ]s ].
(5.4.10)
Proof. Let {⇣k } be stop times such that M k = M [⇣k ] 2 M2 . Then M k is a stochastic
integrator by Theorem 5.24 and thus so is M by Theorem 4.45.
Now given f 2 L2m (M ), let
⌧k = k ^ ⇣k ^ k.
k
" 1 be stop times such that (5.4.8) is true and let
Let gn be bounded predictable processes, with |gn |  |f | such that gn converges to 0
R
pointwise. Let Z n = gn dM . To prove f 2 L(M ), we need to show that ducp (Z n , 0)
converges to 0. In view of Lemma 2.72, suffices to show that for each k 1,
Y n,k = (Z n )[⌧k ]
ucp
!0
as n ! 1.
(5.4.11)
176
Chapter 5. Semimartingales
R
Note that Y n,k = gn 1[0,⌧k ] dM k . Also, Y n,k is a square integrable martingale since gn is
bounded and M k is a square integrable martingale. Moreover,
Z T
n,k
n,k
E([Y , Y ]T ) = E(
(gn )2 1[0,⌧k ] d[M k , M k ])
0
Z ⇣k
 E(
(gn )2 d[M, M ].
0
Sine gn converge pointwise to 0 and |gn |  |f | and f satisfies (5.4.8), it follows that for
each k fixed, E([Y n,k , Y n,k ]T ) ! 0 as n ! 1. Invoking (5.3.22), we thus conclude
lim E[ sup |Ytn,k |2 ] = 0.
n!1
0tT
T
Thus, (5.4.11) holds completing the proof that f 2 L(M ).
The proof that Y is a square integrable martingale and that U is a martingale are
similar to the proof of part (iv) in Theorem 5.16. The estimate (5.4.10) follows invoking
(5.3.24) as Y 2 M2loc .
R
A
F
Remark 5.27. Now that we have shown that locally square integrable r.c.l.l. martingales M
are stochastic integrators, the quadratic variation of M defined via the mapping is consistent
with the definition of [M, M ] given in Theorem 4.59. As a consequence various identities and
inequalities that were proven for the quadratic variation of a stochastic integrator in Section
4.6 also apply to quadratic variation of locally square integrable martingales. Thus from now
on we will drop the superfix in [M, M ] .
D
Remark 5.28. When M is a continuous martinagle with M0 = 0, it follows that M is locally
square integrable (since it is locally bounded). Further, [M, M ]t is continuous and hence for
any predictable f such that
Z t
Dt =
fs2 d[M, M ]s < 1 a.s. ,
(5.4.12)
0
D itself is continuous. Thus D is locally bounded and hence f 2 L2m (M ). It is easy to see
that if f 2 L2m (M ) then f satisfies (5.4.12).
The estimate (5.4.10) has the following implication.
Theorem 5.29. Suppose M n , M 2 M2 are such that
E[ [M n
M, M n
M ]T ] ! 0 8T < 1.
Then M n converges to M in Emery topology.
(5.4.13)
5.5. Semimartingales are Stochastic Integrators
177
Proof. Given predictable f bounded by 1, using (5.4.10) one gets
Z t
Z t
Z T
1
n
P( sup | f dM
f dM | > )  2 E[
|fs |2 d[M n M, M n
0tT
0
0
M ]s ]
0

1
2
E[[M n
M, M n
M ]T ].
This proves M n converges to M in Emery topology in view of the observation (4.9.5).
5.5
Semimartingales are Stochastic Integrators
F
T
In the previous section, we have shown that locally square integrable martingales are
stochastic integrators. In this section, we propose to show that all martingales are integrators and hence by localization it would follow that local martingales are integrators as
well.
Earlier we have shown that processes whose paths have finite variation on [0, T ] for every
T are stochastic integrators. It would then follow that all semimartingales are stochastic
integrators. Here is the continuous analogue of the Burkholder’s inequality, Theorem 1.43.
A
Lemma 5.30. Let Z be a r.c.l.l. martingale with Z0 = 0 and f 2 S1 , namely a simple
predictable process bounded by 1. Then for all > 0, T < 1 we have
Z t
20
P( sup | f dZ| > )  E[|ZT | ].
(5.5.1)
0tT
R
Proof. Let f 2 S1 be given by
0
f (s) = a0 1{0} (s) +
m
X1
aj+1 1(sj ,sj+1 ] (s)
(5.5.2)
j=0
D
where 0 = s0 < s1 < s2 < . . . < sm < 1, aj is Fsj 1 measurable random variable, |aj |  1,
1  j  m and a0 is F0 measurable and |a0 |  1. Without loss of generality, we assume
that sm 1 < T = sm . Then
Z t
m
X
f dZ =
ak (Zsk ^t Zsk 1 ^t ).
(5.5.3)
0
k=1
Let us define a discrete process Mk = Zsk and discrete filtration Gk = Fsk for 0  k  m.
Then M is a martingale with respect to the filtration {Gk : 0  k  m}. Let us also define
Uk = ak , 1  k  m and U0 = 0. Then U is predictable (with respect to {Gk : 0  k  m})
and is bounded by 1 and hence using Theorem 1.43 we conclude that for ↵ > 0,
n
X
9
9
P( max |
Uk (Mk Mk 1 )| ↵)  E[|Mm | ] = E[|ZT | ].
(5.5.4)
1nm
↵
↵
k=1
178
Chapter 5. Semimartingales
Rt
R sk 1
Note that for sk 1  t  sk , defining Vtk = 0 f dZ
f dZ, we have Vtk = ak (Zt
0
Zsk 1 ) and hence
sup |Vtk |  2 sup |Zt |
(5.5.5)
sk
Also, note that
R sk
1 tsk
0tT
P
f dZ = kj=1 Uj (Mj Mj 1 ) and hence
Z t
Z sk
sup | f dZ|  max |
f dZ| + max
0
0tT
1km
0
0
 max |
1km
1km sk
k
X
Uj (Mj
Mj
1 )|
j=1
sup
1 tsk
j=1
+ P(2 sup |Zt | >
1 )|
>
1
)
4
3
)
4
(5.5.7)
F
0tT
0tT
T
1km
0
(5.5.6)
+ 2 sup |Zt |.
Thus using (5.5.4) and (5.5.6) along with Theorem 2.24 we get
Z t
k
X
P( sup | f dZ| > ) P( max | Uj (Mj Mj
0tT
|Vtk |
36
8
E[|ZT | ] + E[|ZT | ]
3
20
= E[|ZT | ].
A

R
Lemma 5.31. Let M be a square integrable r.c.l.l. martingale with M0 = 0 and let g be a
bounded predictable process, |g|  C. Then
Z t
20
P( sup | gdM | > )  C E[|MT | ].
(5.5.8)
0
D
0tT
Proof. When g is simple predictable process, the inequality (5.5.8) follows from Lemma
5.30. Since M is a stochastic integrator, the class of predictable processes g bounded by C
for which (5.5.8) is true is bp- closed and hence it includes all such processes.
Theorem 5.32. Let M be a uniformly integrable r.c.l.l. martingale with M0 = 0. Then
M is a stochastic integrator and (5.5.8) continues to be true for all bounded predictable
process g.
Proof. In view of Theorem 4.30, in order to show that M is a stochastic integrator for the
filtration (F⇧ ), suffices to show that it is a stochastic integrator w.r.t. (F⇧+ ). Recall that M
being r.c.l.l., remains a uniformly integrable martingale w.r.t. (F⇧+ ).
5.5. Semimartingales are Stochastic Integrators
179
Since M is a uniformly integrable martingale, by Theorem 2.23, ⇠ = limt!1 Mt exists
a.e and in L1 (P) and further Mt = E[⇠ | Ft+ ]. For k
1, let M k be the r.c.l.l. (F⇧+ )martingale given by
Mtk = E[⇠ 1{|⇠|k} | Ft+ ]
Note that for any T < 1, and k, j
E[⇠ 1{|⇠|k} | F0+ ].
n
E[|MTk
MTj | ]  2E[|⇠|1{|⇠|>n} ]
(5.5.9)
E[|MTk
MT | ]  2E[|⇠|1{|⇠|>n} ].
(5.5.10)
and
0tT
0
0
F
T
Doob’s maximal inequality - Theorem 2.24 now implies that M k converges to M in ducp
metric.
Let an = E[|⇠|1{|⇠|>n} ]. Since ⇠ is integrable, it follows that an ! 0 as n ! 1.
Since M k is a bounded (F⇧+ )-martingale, it is a square integrable (F⇧+ )-martingale and
hence a (F⇧+ )-stochastic integrator. We will first prove that M k is Cauchy in dem metric.
Note that for any f 2 S1 , using (5.5.8) we have for k, j n,
Z t
Z t
20
40
P( sup | f dM k
f dM j | > )  E[|MTk MTj | ] = an
D
R
A
and hence, using (4.9.6) it follows that {M k : k 1} is Cauchy in dem metric.
Since the class of stochastic integrators is complete in dem metric as seen in Theorem
4.98, and M k converges to M in ducp , it would follow that indeed M is also a (F⇧+ )stochastic integrator and M k converges to M in dem .
R
Since (5.5.8) holds for M k and for any bounded predictable g, gdM k converges to
R
gdM in ducp , it follows that (5.5.8) continues to be true for uniformly integrable martingales M .
The proof given above essentially also contains the proof of the following observations.
Theorem 5.33. Let N, N k , M k for k
(i) If for all T < 1
E[ |NTk
1 be uniformly integrable r.c.l.l. martingales.
NT | ] ! 0 as k ! 1,
then N k converges to N in the Emery topology.
(ii) If for all T < 1
E[ |MTk
MTn | ] ! 0 as k, n ! 1,
then M k is Cauchy in the dem -metric for the Emery topology.
180
Chapter 5. Semimartingales
Here is the final result of this section.
Theorem 5.34. Let an r.c.l.l. process X be a semimartingale, i.e. X can be decomposed
as X = M + A where M is an r.c.l.l. local martingale and A is an r.c.l.l. process with finite
variation paths. Then X is a stochastic integrator.
Proof. We have shown that uniformly integrable r.c.l.l. martingales are stochastic integrators and hence by localization, all r.c.l.l. local martingales are integrators. Thus M is an integrator. Of course, r.c.l.l. processes A with finite variation paths, i.e. Var[0,T ] (A· (!)) < 1
for all T < 1 for all ! 2 ⌦, are integrators and thus X = M + A is a stochastic integrator.
Stochastic Integrators are Semimartingales
T
5.6
F
The aim of this section is to prove the converse to Theorem 5.34. These two results taken
constitute one version of The Bichteler-Dellacherie-Meyer-Mokobodzky Theorem.
Let Z be a stochastic integrator. Let
X
Bt = Z0 +
( Z)s 1{|(
A
0<st
Z)s |>1} .
(5.6.1)
R
Since paths of Z are r.c.l.l., for every !, there only finitely many jumps of Z of size greater
than 1 in [0, t] and thus B is a well defined r.c.l.l. adapted process whose paths are of finite
variation and hence B itself is a stochastic integrator. Thus Y = Z B is a stochastic
integrator and now jumps of Y are of magnitude at most 1. Now defining stop times ⌧n
for n 1 via
D
⌧ n = inf{t > 0 : |Yt |
n or |Yt |
n}
(5.6.2)
n
and Y n = Y [⌧ ] (i.e. Ytn = Yt^⌧ n ), it follows that for each n, Y n is a stochastic integrator
by Lemma 4.43. Further Y n is bounded by n + 1, since its jumps are bounded by 1.
We will show that bounded stochastic integrators X can be decomposed as X = M + A
where M is a r.c.l.l. square integrable martingale and A is an r.c.l.l. process with finite
variation paths. We will also show that this decomposition is unique under a certain condition on A. This would help in piecing together {M n }, {An } obtained in the decomposition
Y n = M n + An of Y n to get a decomposition of Y into an r.c.l.l. locally square integrable
martingale and an r.c.l.l. process with finite variation paths.
The proof of these steps is split into several lemmas.
5.6. Stochastic Integrators are Semimartingales
181
Lemma 5.35. Let M n 2 M2 be a sequence of r.c.l.l. square integrable martingales such
n
that M0n = 0. Suppose 9 T < 1 such that Mtn = Mt^T
and
E[[M n
M k, M n
M k ]T ] ! 0 as min(k, n) ! 1.
(5.6.3)
Then there exists an r.c.l.l. square integrable martingale M 2 M2 such that
lim E[[M n
M, M n
n!1
dem (M n , M ) ! 0 and
lim E[ sup |Mtn
n!1
0tT
M ]T ] = 0,
Mt |2 ] = 0.
(5.6.4)
(5.6.5)
Proof. The relation (5.3.23) and the hypothesis (5.6.3) implies that
E[ sup |Msn
Msk |2 ] ! 0 as min(k, n) ! 1.
(5.6.6)
T
0tT
A
F
Hence the sequence of processes {M n } is Cauchy in ucp metric and thus in view of Theorem
2.68 converges to an r.c.l.l. adapted process M and (5.6.6) is satisfied. Further, (5.6.6) also
implies that Msn converges to Ms in L2 (P) for each s and hence using Theorem 2.21 it
follows that M is a martingale, indeed a square integrable martingale. As a consequence
(MTn MT ) ! 0 in L2 (P) and using (5.3.23) it follows that (5.6.4) is true. The convergence
of M n to M in Emery topology follows from Theorem 5.29.
If M n in the Lemma above are continuous, it follows that M is also continuous. This
gives us the following.
R
Corollary 5.36. Let M n 2 M2 be a sequence of continuous square integrable martingales
n
such that M0n = 0. Suppose 9 T < 1 such that Mtn = Mt^T
and
D
E[[M n
M k, M n
M k ]T ] ! 0 as min(k, n) ! 1.
(5.6.7)
Then there exists a continuous square integrable martingale M 2 M2 such that
lim E[[M n
n!1
dem (M n , M ) ! 0 and
M, M n
lim E[ sup |Mtn
n!1
0tT
M ]T ] = 0
Mt |2 ] = 0.
Theorem 5.37. Let X be a stochastic integrator such that
(i) Xt = Xt^T for all t,
(ii) E[[X, X]T ] < 1.
(5.6.8)
(5.6.9)
182
Chapter 5. Semimartingales
Then X admits a decomposition X = M + A where M is an r.c.l.l. square integrable
martingale and A is a stochastic integrator satisfying
E[[N, A]T ] = 0
(5.6.10)
for all r.c.l.l. square integrable martingales N .
Proof. The proof is very similar to the proof of existence of the projection operator on a
Hilbert space onto a closed subspace of the Hilbert space. Let
↵ = inf{E[[X
M ]T ] : M 2 M2 }.
M, X
1, let M̃ k 2 M2 be such that
Since E[[X, X]T ] < 1, it follows that ↵ < 1. For k
k , t
Define Mtk = M̃t^T
1
M̃ k ]T ]  ↵ + .
k
1. Then
M k, X
M k ]T ] = E[[X
1
M̃ k ]T ]  ↵ + .
k
M̃ k , X
F
E[[X
0, k
M̃ k , X
T
E[[X
Applying the parallelogram identity (4.6.13) to Y k = 12 (X
we get
Y n, Y k
Y n ]T = 2[Y k , Y k ]T + 2[Y n , Y n ]T
A
[Y k
1
n
2 (M
Note that Y k + Y n = X
Since Y k
R
Y n = 12 (M n
(5.6.11)
↵
1
1
1
1
Y n ]T ]  2( (↵ + )) + 2( (↵ + ))
4
k
4
n
Y n, Y k
D
E[[Y k
[Y k + Y n , Y k + Y n ]T
M n)
+ M k ) and since 12 (M n + M k ) 2 M2 , we have
E[[Y k + Y n , Y k + Y n ]T ]
and hence
M k ), Y n = 21 (X
↵
(5.6.12)
M k ), (5.6.11)-(5.6.12) yields
1
E[[M n
4
M k, M n
1 1
1
M k ]T ]  ( + )
2 k n
Thus by Lemma 5.35, it follows that there exists M 2 M2 such that
lim E[[M n
n!1
M, M n
M ]T ] = 0.
We now show that ↵ is attained for this M . Let us define Y = 12 (X
Y n Y = 12 (M M n ) and hence (5.6.13) yields
E[[Y n
Y, Y n
Y ]T ] ! 0.
(5.6.13)
M ). Then we have
(5.6.14)
5.6. Stochastic Integrators are Semimartingales
183
Using (4.6.14) we have
|E[[Y n , Y n ]T ]
E[[Y, Y ]T ]|
 E[|[Y n , Y n ]T [Y, Y ]T | ]
p
 E[ 2([Y n Y, Y n Y ]T ).([Y n , Y n ]T + [Y, Y ]T )]
p
 2(E[[Y n Y, Y n Y ]T ])(E[[Y n , Y n ]T + [Y, Y ]T ])
(5.6.15)
Since E[[Y n , Y n ]T ]  14 (↵ + n1 ) and E[[Y, Y ]T ]  2(E[[X, X]T + [M, M ]T ]) < 1, (5.6.14) and
(5.6.15) together yield E[[Y n , Y n ]T ] ! E[[Y, Y ]T ] as n ! 1 and hence E[[Y, Y ]T ]  14 ↵.
1
On the other hand, since Y = 12 (X M ) where M 2 M2 , we have E[[Y, Y ]T ]
4 ↵ and
1
1
hence E[[Y, Y ]T ] = 4 ↵. Recalling Y = 2 (X M ) we conclude E[[X M, X M ]T ] = ↵.
By definition of ↵ we have, for any N 2 M2 , for all u 2 R
M
uN, X
M
↵ = E[[X
uN ]T ]
since M + uN 2 M2 . We thus have
2uE[[N, X
M ]T ]
0 for all u 2 R.
M ]T ]
(5.6.16)
F
u2 E[[N, N ]T ]
M, X
T
E[[X
This implies E[[N, X M ]T ] = 0. Now the result follows by setting A = X M . A is a
stochastic integrator because X is so by hypothesis and M has been proven to be so.
A
Recall that M2c denotes the class of continuous square integrable martingales. A small
modification of the proof above yields the following.
R
Theorem 5.38. Let X be a stochastic integrator such that
(i) Xt = Xt^T for all t,
D
(ii) E[[X, X]T ] < 1.
Then X admits a decomposition X = N + Y where N is a continuous square integrable
martingale and Y is a stochastic integrator satisfying
E[[Y, U ]T ] = 0 8U 2 M2c .
(5.6.17)
Proof. This time we define
↵0 = inf{E[[X
M, X
M ]T ] : M 2 M2c }
and proceed as in the proof of Theorem 5.37.
The next lemma shows that (5.6.10) implies an apparently stronger conclusion- that
[N, A] is a martingale for N 2 M2 .
184
Chapter 5. Semimartingales
Lemma 5.39. Let A be a stochastic integrator such that
(i) At = At^T , 8t < 1,
(ii) E[[A, A]T ] < 1,
(iii) E[[N, A]T ] = 0 for all N 2 M2 .
Then [N, A] is a martingale for all N 2 M2 .
T
Proof. Fix N 2 M2 . For any stop time , N [ ] is also a square integrable martingale and
[ ]
(4.6.9) gives [N [ ] , A]T = [N, A]T = [N, A]T ^ and thus we conclude E[[N, A]T ^ ] = 0.
Theorem 2.53 now implies that [N, A] is a martingale.
F
Remark 5.40. If U is a continuous square integrable martingale and is a stop time, U [ ]
is also a continuous square integrable martingale. Hence, arguments as in the proof of the
previous result yields that if a stochastic integrator Y satisfies (5.6.17), then
[Y, U ] is a martingale 8U 2 M2c .
A
The next result would tell us that essentially, the integrator A obtained above has finite
variation paths (under some additional conditions).
R
Lemma 5.41. Let A be a stochastic integrator such that
(i) At = At^T , 8t < 1,
D
(ii) E[B] < 1 where B = suptT |At | + [A, A]T ,
(iii) [N, A] is a martingale for all N 2 M2 (F⇧+ ).
Then A is a process with finite variation paths: Var[0,T ] (A) < 1 a.s.
Proof. For a partition ⇡
˜ = {0 = s0 < s1 < . . . < sm = T } of [0, T ] let us denote
V ⇡˜ =
m
X
j=1
|Asj
Asj 1 |.
We will show that for all " > 0, 9K < 1 such that
sup P(V ⇡
⇡
K) < "
(5.6.18)
5.6. Stochastic Integrators are Semimartingales
185
where the supremum above is taken over all partitions of [0, T ]. Taking a sequence ⇡ n of
successively finer partitions such that (⇡ n ) ! 0 (for example, ⇡ n = {kT 2 n : 0  k 
n
2n }), it follows that V ⇡ " Var[0,T ] (A) and thus (5.6.18) would imply
P(Var[0,T ] (A)
K) < "
and hence that P(Var[0,T ] (A) < 1) = 1. This would complete the proof.
Fix " > 0. Since A is a stochastic integrator for the filtration (F⇧ ), it is also a stochastic
integrator for the filtration (F⇧+ ) in view of Theorem 4.3. Thus we can get (see Remark
4.23) J1 < 1 such that
Z T
"
sup P(|
f dA| J1 )  .
(5.6.19)
6
0
f 2S1 (F + )
⇧
Since E[B] < 1, we can get J2 < 1 such that
"
J2 )  .
(5.6.20)
6
Let J = max{J1 , J2 , E[B]} and n be such that 24
n < ". Let K = (n + 1)J. We will show
that (5.6.18) holds for this choice of K. Note that the choice has been made independent
of a partition.
Now fix a partition ⇡ = {0 = t0 < t1 < . . . < tm = T }. Recall that for x 2 R,
sgn(x) = 1 for x 0 and sgn(x) = 1 for x < 0, so that |x| = sgn(x)x. For 1  j  m, let
us consider the (F⇧+ )-martingale
A
F
T
P(B
Z̃tj = E[sgn(Atj
Atj 1 ) | Ft+ ].
R
Since the filtration (F⇧+ ) is right continuous, the martingale Z̃tj admits an r.c.l.l. version
Ztj . Then Ztjj = sgn(Atj Atj 1 ) and hence
1(tj
Atj 1 ) = |(Atj
dAs = (At^tj
|(Atj
0
1 ,tj ]
Atj 1 )| = Ztjj Ctjj
Z tj
Z tj
j
j
=
Zs dCs +
Csj dZsj + [Z j , C j ]tj
0
0
Z tj
Z tj
j
=
Zs dAs +
Csj dZsj + [Z j , A]tj
tj
and for tj
1
Atj 1 )|.
At^tj 1 ), we get by integration by parts formula
D
Writing Ctj =
(4.6.7)
Rt
Ztjj (Atj
 t  tj
Ztj Ctj
=
Z
t
tj
Zsj
1
tj
1
dAs +
Z
t
tj
1
Csj dZsj + [Z j , A]t
1
[Z j , A]tj
[Z j , A]tj 1 .
1
186
Chapter 5. Semimartingales
Let us define
Zt =
m
X
Ztj 1(tj
1 ,tj ]
(t),
Ctj 1(tj
1 ,tj ]
(t),
j=1
Ct =
m
X
j=1
Mt =
m
X
j
(Zt^t
j
j
Zt^t
).
j 1
j=1
It follows that |Cs |  2B, |Zs |  1, M is a bounded (F⇧+ )- martingale and
m
X
[M, A]t =
([Z j , A]t^tj [Z j , A]t^tj 1 ).
Yt =
we get, for tk
1
Z
t
Zs dAs +
0
 t < tk , 1  k  m
Yt =
k 1
X
j=1
t
Cs dMs + [M, A]t
0
Atj 1 )| + Ztk Ctk
2B. Also note that
A
and thus Yt
|(Atj
Z
F
Thus, defining
T
j=1
YT =
Rt
m
X
j=1
Atj 1 )| = V ⇡ .
|(Atj
D
R
Let Ut = 0 Cs dMs + [M, A]t , it follows that U is a (F⇧+ )-local martingale since by
Rt
assumption on A, [M, A] is itself a (F⇧+ )-martingale and thus 0 Cs dMs is a (F⇧+ )local
martingale by Theorem 5.26. Further
Z t
Yt =
Zs dAs + Ut .
0
Now defining
⌧ = inf{t
0 : Ut <
3J} ^ T
it follows that ⌧ is a stop time (see Lemma 2.46) and
{⌧ < T } ✓ {U⌧ 
Since Y⌧
3J}.
2B, we note that
({⌧ < T } \ {B < J}) ✓ ({U⌧  3J} \ {Y⌧
Z ⌧
✓{
Zs dAs J}.
0
2J}
5.6. Stochastic Integrators are Semimartingales
Hence using (5.6.19) and (5.6.20) we conclude
Z ⌧
P(⌧ < T )  P(
Zs dAs
187
"
J)  .
(5.6.21)
3
0
Now ( U )t = Ct ( M )t + ( M )t ( A)t . Since M is bounded by 1, ( M )t  2. Also,
|Ct |  2B and |( A)t |  2B. Thus |( U )t |  8B. Let n be (F⇧+ )-stop times such that
U [ n ] is a (F⇧+ )-martingale. By definition of ⌧ , it follows that Ut^ n ^⌧
(3J + 8B). Since
+
[
]
n
U
is a (F⇧ )-martingale, we have E[UT ^ n ^⌧ ] = 0. Recall that B is integrable and hence
using Fatou’s lemma we conclude
E[UT ^⌧ ]  0.
(5.6.22)
Since UT ^⌧
(3J + 8B) and J
using (5.6.22) we conclude
0
E[B], it follows that E[(UT ^⌧ ) ]  E[(3J + 8B)]. Thus
E[(UT ^⌧ )+ ]  11J.
(5.6.23)
T
Since V ⇡ = YT =
RT
J) + P(B
Zs dAs + UT and recalling that K = (n + 1)J
Z T
⇡
P(V
K)  P(
Zs dAs J) + P(UT nJ)
0
since by our choice
24
n
R
A
F
"
 + P(⌧ < T ) + P(U⌧ nJ)
6
" "
1
 + +
E[(U⌧ )+ ]
6 3 nJ
" 11J
 +
2
nJ
" "
< + ="
2 2
< ". This proves (5.6.18) and completes the proof as noted earlier.
D
We now put together the results obtained earlier in this chapter to get the following
key step in the main theorem of the section. We have to avoid assuming right continuity
of the filtration in the main theorem. However, it required in the previous result and we
avoid the same by an interesting argument. Here is a lemma that is useful here and in
later chapter.
Lemma 5.42. Let (⌦, F, P) be a complete probability space and let H ✓ G be sub- fields
of F such that H contains all the P null sets in F. Let Z be an integrable G measurable
random variable such that for all G measurable bounded random variables U one has
E[ZU ] = E[ZE[U | H]].
Then Z is H measurable.
(5.6.24)
188
Chapter 5. Semimartingales
Proof. Noting that
E[ZE[U | H]] = E[E[Z | H]E[U | H]] = E[E[Z | H]U ]
using (5.6.24) it follows that for all G measurable bounded random variables U
E[(Z
E[Z | H])U ] = 0.
Taking U = sgn(Z E[Z | H]) (here sgn(x) = 1 for x
conclude from (5.6.25) that
E[ |(Z
0 and sgn(x) =
(5.6.25)
1 for x < 0), we
E[Z | H])| ] = 0.
Since H is assumed to contain all P null sets in F, it follows that Z is H measurable.
T
Theorem 5.43. Let X be a stochastic integrator such that
(i) Xt = Xt^T for all t,
F
(ii) E[supsT |Xs | ] < 1,
(iii) E[[X, X]T ] < 1.
A
Then X admits a decomposition
X = M + A, M 2 M2 , A 2 V,
(5.6.26)
[N, A] is a martingale for all N 2 M2
(5.6.27)
E[[X, X]T ] = E[[M, M ]T ] + E[[A, A]T ]
(5.6.28)
D
and
R
such that
and further, the decomposition (5.6.26) is unique under the requirement (5.6.27).
Proof. As seen in Theorem 4.30, X being a stochastic integrator for the filtration (F⇧ )
implies that X is also a stochastic integrator for the filtration (F⇧+ ). Also, (4.6.2) implies
that [X, X] does not depend upon the underlying filtration, so the assumptions of the
Theorem continue to be true when we take the underlying filtration to be (F⇧+ ). Now
Theorem 5.37 yields a decomposition X = M̃ + Ã with M̃ 2 M2 (F⇧+ ) and E[[N, Ã]T ] = 0
for all N 2 M 2 (F⇧+ ). Let Mt = M̃t^T , At = Ãt^T . Then M 2 M2 (F⇧+ ) and E[[N, A]T ] = 0
for all N 2 M 2 (F⇧+ ) since [N, A]t = [N, Ã]t^T (by (4.6.9)). As a consequence,
E[[X, X]T ] = E[[M, M ]T ] + E[[A, A]T ].
5.6. Stochastic Integrators are Semimartingales
189
Thus E[[A, A]T ] < 1 and then by Lemma 5.39, we have [N, A] is a (F⇧+ )-martingale
for all N 2 M2 (F⇧+ ). Since the underlying filtration (F⇧+ ) is right continuous, Lemma 5.41
implies that A 2 V, namely paths of A have finite variation. By construction, M, A are
(F⇧+ ) adapted. Since for s < t, Fs+ ✓ Ft , it follows that At = At is Ft measurable. We
will show that
At is Ft measurable 8t > 0.
(5.6.29)
Since Xt is Ft measurable, it would follow that Mt = Xt At is also Ft measurable. This
will also imply M 2 M2 = M2 (F⇧ ) completing the proof.
Fix t > 0. Let U be a bounded Ft+ measurable random variable and let V = U E[U |
Ft ]. Let
Ns = V 1[t,1) (s)
T
i.e. Ns = 0 for s < t and Ns = V for s
t. It is easy to see that N 2 M2 (F⇧+ ) and
that [N, A]s = V ( A)t 1[t,1) (s) (using (4.6.10)). Thus [N, A] is a (F⇧+ )-martingale, and in
particular E[[N, A]t ] = 0. Hence we conclude that for all bounded Ft+ measurable U
F
E[( A)t U ] = E[( A)t E[U | Ft ]].
(5.6.30)
D
R
A
Invoking Lemma 5.42 we conclude that ( A)t is Ft measurable and hence that At is Ft
measurable. As noted earlier, this implies M 2 M2 . Only remains to prove uniqueness of
decomposition satisfying 5.6.27. Let X = Z + B be another decomposition with Z 2 M2 ,
B 2 V and [N, B] being a martingale for all N 2 M2 .
Now X = M + A = Z + B, B A = M Z 2 M2 and [N, B A] is a martingale
for all N 2 M2 . Let N = M
Z = B A. By definition, M0 = Z0 = 0 and so
N0 = 0. Now [N, B A] is a martingale implies [B A, B A] = [M Z, M Z] is a
martingale and as a consequence, we have E[[M Z, M Z]T ] = E[[M Z, M Z]0 ] = 0
(recall [M Z, M Z]0 = 0, see Remark 4.66). Now invoking (5.3.22), we conclude (since
M Z 2 M2 and M0 = Z0 = 0)
E[sup|Ms
sT
Zs |] = 0.
Thus M = Z and as a consequence A = B. This completes the proof.
Corollary 5.44. The processes M, A in (5.6.26) satisfy
Mt = Mt^T ,
At = At^T 8t
0
(5.6.31)
Proof. Note that Xt = Xt^T . Let Rt = Mt^T and Bt = At^T . Then X = R + B is also a
decomposition that satisfies (5.6.26) since R is also a square integrable martingale, B is a
190
Chapter 5. Semimartingales
process with finite variation paths and if R is any square integrable martingale, then
[N, B]t = [N, A]t^T
and hence [N, B] is also a martingale. Now uniqueness part of Theorem 5.43 implies
(5.6.31).
Corollary 5.45. Suppose X, Y are stochastic integrators satisfying conditions of Theorem 5.43 and let X = M +A and Y = N +B be decompositions with M, N 2 M2 , A, B 2 V
and [U, A], [U, B] being martingales for all U 2 M2 . If for a stop time , X [ ] = Y [ ] , then
M [ ] = N [ ] and A[ ] = B [ ] .
We now introduce two important definitions.
T
Proof. Follows by observing that X [ ] is also a stochastic integrator and X [ ] = M [ ] + A[ ]
and X [ ] = N [ ] + B [ ] are two decompositions, both satisfying (5.6.27). The conclusion
follows from the uniqueness part of Theorem 5.43.
F
Definition 5.46. An adapted process B is said to be locally integrable if there exist stop
times ⌧n increasing to 1 and random variables Dn such that E[Dn ] < 1 and
P(! :
sup
|Bt (!)|  Dn (!)) = 1 8n
1.
A
0t⌧n (!)
R
The condition above is meaningful even if sup0t⌧n (!) |Bt (!)| is not measurable. It
is to be interpreted as - there exists a set ⌦0 2 F with P(⌦0 ) = 1 such that the above
inequality holds for ! 2 ⌦0 .
D
Definition 5.47. An adapted process B is said to be locally square integrable if there exist
stop times ⌧n increasing to 1 and random variables Dn such that E[Dn2 ] < 1 and
P(! :
sup
0t⌧n (!)
|Bt (!)|  Dn (!)) = 1 8n
1.
Clearly, if B is locally bounded process then it is locally square integrable. Easy to
see that a continuous adapted processes Y is locally bounded if Y0 is bounded, is locally
integrable if E[ |Y0 | ] < 1 and locally square integrable if E[ |Y0 |2 ] < 1. Indeed, the same
is true for an r.c.l.l. adapted process if its jumps are bounded.
We have seen that for an r.c.l.l. adapted process Z, the process Z is locally bounded
and hence it follows that Z is locally integrable if and only if the process Z is locally
integrable and likewise, Z is locally square integrable if and only if the process Z is locally
square integrable.
5.6. Stochastic Integrators are Semimartingales
191
Theorem 5.48. Let X be locally square integrable stochastic integrator. Then X admits a
decomposition X = M + A where M 2 M2loc (M is a locally square integrable martingale)
and A 2 V (A is a process with finite variation paths) satisfying
[A, N ] 2 M2loc 8N 2 M2loc .
(5.6.32)
Further, such a decomposition is unique.
Proof. Since X is a locally square integrable process, it follows that X is locally square
integrable. Since [X, X] = ( X)2 , it follows that [X, X] is locally integrable and thus so
is Dt = supst |Xs |2 + [X, X]t . Let n " 1 be stop times such that E[D n ] < 1 and let
n
⌧ n = n ^ n . Let X n = X [⌧ ] . Then X n satisfies conditions of Theorem 5.43 (with T = n)
and thus we can get decomposition X n = M n + An such that M n 2 M2 . An 2 V and
[U, An ] is a martingale for all U 2 M2 .
T
(5.6.33)
Using Corollary 5.45, we can see that
P(Mtn = Mtk 8t  ⌧ n ^ ⌧ k ) = 1, 8n, k.
n
n
A
F
Thus we can define r.c.l.l. processes M, A such that M [⌧ ] = M n and A[⌧ ] = An . This
decomposition satisfies the asserted properties.
Uniqueness follows as in Theorem 5.43 and the observation that if Y 2 M2loc , Y0 = 0
and [Y, Y ]t = 0 for all t then Y = 0 (i.e. P(Yt = 0 8t) = 1.)
R
Remark 5.49. The process A with finite variation r.c.l.l. paths appearing in the above theorem
was called a Natural process by Meyer and it appeared in the Doob Meyer decomposition of
supermartingales. Later it was shown that such a process is indeed a predictable process. A is
also known as the compensator of X. We will come back to this in Chapter 8 later
D
Corollary 5.50. Let X be a locally square integrable stochastic integrator and A be its
compensator and M = X A 2 M2loc . Then for any stop time such that E[ [X, X] ] < 1,
E[ [A, A] ]  E[ [X, X] ]
(5.6.34)
E[ [M, M ] ]  E[ [X, X] ]
(5.6.35)
and
Proof. If ⌧n " 1 are as in the proof of Theorem 5.48, then by Theorem 5.43 we have
E[[X, X]
^⌧n ]
= E[[M, M ]
^⌧n ]
+ E[[A, A]
^⌧n ].
and the required inequalities folow by taking limit as n ! 1 and using monotone convergence theorem.
192
Chapter 5. Semimartingales
Arguments similar to the ones leading to Theorem 5.48 yield the following (we use
Theorem 5.38 and Remark 5.40). We also use the fact that every continuous process is
locally bounded and hence locally square integrable.
Theorem 5.51. Let X be locally square integrable stochastic integrator. Then X admits
a decomposition X = N + Y with N 2 Mc,loc ( N is a continuous locally square integrable
martingale with N0 = 0) and Y is a locally square integrable stochastic integrator satisfying
[Y, U ] is a local martingale 8U 2 M2c .
(5.6.36)
Further, such a decomposition is unique and
[Y, U ] = 0 8U 2 M2c .
(5.6.37)
F
T
Proof. The only new part is to show that (5.6.36) yields (5.6.37). For this note that on
one hand [Y, U ] is continuous as [Y, U ] = ( Y )( U ) and U is continuous and on the
other hand by definition [Y, U ] has finite variation paths. Thus [Y, U ] is a continuous local
martingale with finite variation paths. Hence by Theorem 5.19, [Y, U ] = 0.
A
As an immediate consequence of the Theorem 5.48, here is a version of the BichtelerDellacherie-Meyer-Mokobodzky Theorem. See Theorem 5.78 for the final version.
R
Theorem 5.52. Let X be an r.c.l.l. adapted process. Then X is a stochastic integrator if
and only if X is a semimartingale.
Proof. We have already proved (in Theorem 5.34) that if X is a semimartingale then X is
a stochastic integrator.
D
For the other part, let X be a stochastic integrator. Let us define
X
Bt = X0 +
( X)s 1{|( X)s |>1} .
(5.6.38)
0<st
Then as noted at the beginning of the section, B is an adapted r.c.l.l. process with finite
variation paths and is thus a stochastic integrator. Hence Z = X B is also a stochastic
integrator. By definition,
( Z) = ( X)1{|
X|1}
and hence jumps of Z are bounded by 1. Hence Z is locally square integrable. Hence by
Theorem 5.48, Z admits a decomposition Z = M + A where M 2 M2loc and A is a process
with finite variation paths. Thus X = M + (B + A) and thus X is a semimartingale.
5.6. Stochastic Integrators are Semimartingales
193
The result proven above contains a proof of the following fact, which we record here
for later reference.
Corollary 5.53. Every semimartingale X can be written as X = M +A where M 2 M2loc
and A 2 V i.e. M is a locally square integrable r.c.l.l. martingale with M0 = 0 and A is an
r.c.l.l. process with finite variation paths.
Every local martingale X is a semimartingale by definition. In that case, the process
A appearing above is also a local martingale. Thus we have
T
Corollary 5.54. Every r.c.l.l. local martingale N can be written as N = M + L where
M is a locally square integrable r.c.l.l. martingale with M0 = 0 and L is an r.c.l.l. process
with finite variation paths that is also a local martingale i.e. M 2 M2loc and L 2 V \ Mloc .
F
Using the technique of separating large jumps from a semimartingale to get a locally
bounded semimartingale used in proof of Theorem 5.78, we can get the following extension
of Theorem 5.51.
A
Theorem 5.55. Let X be a stochastic integrator. Then X admits a decomposition X =
N +S with N 2 Mc,loc ) ( N is a continuous locally square integrable martingale with N0 = 0)
and S is a stochastic integrator satisfying
(5.6.39)
R
[S, U ] = 0 8U 2 M2c .
Further, such a decomposition X = N + S is unique.
D
Proof. Let B be defined by (5.6.38) and let Z = X B. Then Z is locally bounded and
thus invoking Theorem 5.51, we can decompose Z as Z = N + Y with N 2 Mc,loc and Y
satisfying [Y, U ] = 0 for all U 2 M2c . Let S = Y + B. Since [B, U ] = 0 for all U 2 M2c , it
follows that X = N + S and [S, U ] = 0 for all U 2 M2c . To see that such a decomposition
is unique, if X = M + R is another such decomposition with M M2c and [R, U ] = 0 for all
U 2 M2c , then V = N M = R S 2 Mc,loc with [V, V ] = 0 and hence V = 0 by Theorem
5.19.
Definition 5.56. Let X be a semimartingale. The continuous local martingale N such that
X = N + S and [S, U ] = 0 for all U 2 Mc,loc is said to be the continuous local martingale
part of X and is denoted by X (c) .
194
Chapter 5. Semimartingales
For a semimartingale X with X = X (c) + Z, we can see that [X, X] = [X (c) , X (c) ] +
[Z, Z]. We will later show in Theorem 8.76 that indeed [X (c) , X (c) ] = [X, X](c) - the
continuous part of the quadratic variation of X.
The next result shows that if X is a continuous process that is a semimartingale, then
it can be uniquely decomposed as a sum of a continuous local martingale and a continuous
process with finite variation paths.
Theorem 5.57. Let X be a continuous process and further let X be a semimartingale.
Then X can be uniquely decomposed as
X =M +A
T
where M and A are continuous processes, M0 = 0, M a local martingale and A a process
with finite variation paths.
F
Proof. Without loss of generality, we assume X0 = 0. Now X being continuous, it follows
that X is locally square integrable and thus X admits a decomposition X = M + A with
M 2 M2loc and A 2 V with A satisfying (5.6.32). On one hand, continuity of X implies
( M )t = ( A)t for all t > 0 and since A 2 V, we have
X
[A, M ]t =
( A)s ( M )s .
A
0<st
Thus,
[A, M ]t =
X
( A)2s =
[A, A]t .
0<st
D
R
Since A satisfies (5.6.32), it follows that [A, M ] = [A, A] is a local martingale. If n is a
localizing sequence, it follows that E[[A, A]t^ n ] = 0 for all n. Since [A, A]s 0 for all s, it
follows that [A, A]t^ n = 0 a.s. for all t, n. This implies
X
[A, A]t =
( A)2s = 0 a.s. 8t
0<st
and hence A is a continuous process and hence so is M = X
Theorem 5.19.
A. Uniqueness follows from
Exercise 5.58. Let X be a continuous semimartingale and let X = M +A be a decomposition
as in Theorem 5.57 with M 2 Mc,loc with M0 = 0 and A 2 V. Let X = N + B be any
decomposition with N 2 Mloc and B 2 V. Then [N, N ] [M, M ] is an increasing process, i.e.
[N, N ] = [M, M ] + C,
Hint: Observe that [M, A
C = [A B, A B].
for some C 2 V+ .
B] = 0 since M is continuous. Write N = M + (A
(5.6.40)
B) and take
5.7. The Class L(X)
195
We have earlier introduced the class Mc of continuous martingales with M0 = 0. Let
Md denote the class of martingales N 2 M such that
[M, N ] = 0 8t, 8M 2 Mc .
(5.6.41)
Md is the class of purely discontinuous martingales. Likewise, we define Md,loc to be the
class of M 2 Mloc such that (5.6.41) is true for all N 2 Mc,loc and such M are called purely
discontinuous local martingales. Also, Md,loc \ M2loc is denoted by M2d,loc . We will now
show that every local martingale can be (uniquely) decomposed as a sum of a continuous
local martingale and a purely discontinuous local martingale.
Theorem 5.59. Let X be an r.c.l.l. local martingale. Then X admits a decomposition
Proof. Let
Bt = X0 +
X
(5.6.42)
T
X = M + N, M 2 Mc,loc , N 2 Md,loc .
( X)s 1{|(
X)s |>1}
F
0<st
A
and let Y = X B. Then B is an adapted r.c.l.l. process with finite variation paths and Y
is a stochastic integrator with jumps bounded by 1 and hence is locally square integrable.
Then invoking Theorem 5.51, we get a decomposition
Y =M +A
D
R
where M 2 Mc,loc and A is a locally square integrable stochastic integrator such that
[A, S]t = 0 for all S 2 M2c . Since B 2 V, it follows that [B, S] = 0 for all S 2 M2c . Defining
N = A + B = X M , we get that [N, S] = 0 for all S 2 M2c and since X, M are local
martingales, it follows that so is N .
Remark 5.60. Here, if X 2 M2loc then M 2 M2c,loc and N 2 M2d,loc . We will later show in
Theorem 8.73 that in this case [M, M ] is the continuous part of [X, X] and [N, N ] is the sum
of squares of jumps of X.
5.7
The Class L(X)
In the previous section, we have given a characterization of stochastic integrators. Like
R
stochastic integrators, the class L(X) of integrands for the integral f dX had been defined
in an ad-hoc fashion. Here we give a concrete description of L(X).
196
Chapter 5. Semimartingales
Theorem 5.61. Let X be a stochastic integrator. Then a predictable process f belongs
to L(X) if and only if X admits a decomposition X = M + A, where M 2 M2loc (M is a
locally square integrable martingale with M0 = 0) and A 2 V (A is a process with finite
variation paths) such that
Z t
|fs |d|A|s < 1 8t < 1 a.s.
(5.7.1)
0
" 1 such that
Z k
E[
|fs |2 d[M, M ]s ] < 1 8t < 1.
and there exist stop times
k
(5.7.2)
0
A
F
T
Proof. If f satisfies (5.7.1), then as seen in Remark 4.22 f 2 L(A) and if f satisfies (5.7.2),
then f 2 L2m (M ) ✓ L(M ). Thus if X admits a decomposition X = M + A such that
(5.7.1)-(5.7.2) holds then f 2 L(M ) \ L(A) ✓ L(M + A).
Conversely, let f 2 L(X). Then easy to see that h = (1 + |f |) 2 L(X). So let
R
Y = (1 + |f |)dX. Then as noted in Theorem 4.29, Y is a stochastic integrator. Hence by
Corollary 5.53, Y admits a decomposition Y = N + B and with N 2 M2loc and B 2 V. Let
Z
M = (1 + |f |) 1 dN,
Z
A = (1 + |f |) 1 dB.
R
The two integrals are defined as (1 + |f |) 1 is bounded. Further, this also yields, M 2 M2loc
and A 2 V. Clearly
Z
Z
M + A = (1 + |f |) 1 dY = (1 + |f |) 1 (1 + |f |)dX = X.
1
is bounded, g 2 L(Y ) and hence f = g · (1 + |f |) 2 L(X) (see
D
Since g = f · (1 + |f |)
Theorem 4.29).
Exercise 5.62. Let M be a continuous martingale with M0 = 0.
(i) Let M = N + B be any decomposition with N 2 M2loc and B 2 V and let f 2 L2m (N ).
Show that f 2 L2m (M ).
Hint: Use (5.6.40).
(ii) Show that L(M ) = L2m (M ).
(iii) Show that
L(M ) = {f : predictable such that
Hint: Use Remark 5.28.
Z
t
0
fs2 d[M, M ]s < 1}.
(5.7.3)
5.7. The Class L(X)
197
Remark 5.63. It follows that for a Brownian motion B,
Z t
L(B) = {f : predictable such that
fs2 ds < 1}.
0
Definition 5.64. For a process A with finite variation paths, let L1l (A) be the class of
predictable processes f satisfying (5.7.1).
Thus, L1l (A) ✓ L(A). The Theorem 5.61 can be recast as: For a stochastic integrator
X, f 2 L(X) if and only if X admits a decomposition X = M + A, where M is a locally
square integrable martingale with M0 = 0 and A is a process with finite variation paths
such that
f 2 L2m (M ), and f 2 L1l (A).
{M,A :X=M +A, M 2M2loc , A2V}
L2m (M ) \ L1l (A) .
T
As a consequence, for a semimartingale X
[
L(X) =
F
Exercise 5.65. Let X be a continuous semimartingale and let X = M + A be the unique
decomposition with M being continuous local martingale and A 2 V. Then show that
L(X) = L2m (M ) \ L1l (A).
A
Hint: To show , L(X) ✓ L2m (M ) \ L1l (A) use Exercise 5.58.
The following exercise gives an example of a process A 2 V such that L(A) is strictly
larger than L1l (A).
R
Exercise 5.66. Let {⇠ k,m : 1  k  2m
distributed random variables with
1, m
1} be a family of independent identically
P(⇠ 1,1 = 1) = P(⇠ 1,1 =
2k 1
2m .
Let
D
and let ak,m =
1) = 0.5
Ft = {⇠ k,m : ak,m  t},
m 1
Ant
=
n 2X
X
m=1 k=1
m 1
At =
1 2X
X
m=1 k=1
and f : [0, 1) 7! [0, 1) be defined by
with f (t) = 0 otherwise. Show that
1
22m
1
22m
⇠ k,m 1[an,m ,1) (t),
⇠ k,m 1[an,m ,1) (t)
f (ak,m ) = 2m
198
Chapter 5. Semimartingales
(i) For each n, (Ant , Ft ) is a martingale.
(ii) For each t, Ant converges to At in L2 (P).
(iii) (At , Ft ) is a martingale.
(iv) A 2 V.
(v) Let Bt = |A|t . Show that
(vi) Show that [A, A]t =
5.8
Rt
0
0
f (s)dBs = 1.
P 2m
k=1
1
1
1 n,m ,1) (t).
24m [a
f 2 (s)d[A, A]s < 1.
f dA is defined as a stochastic integral but not defined as a Riemann-Stieltjes integral.
The Bichteler-Dellacherie-Meyer-Mokobodzky Theorem
F
Thus
R1
m=1
0
T
(vii) Show that
P1
Rt
R
A
In Theorem 5.52 we have shown that an r.c.l.l. adapted process is a stochastic integrator if
and only if it is a semimartingale. Even if we demand seemingly weaker requirements on
an adapted r.c.l.l. process X it implies that it is a semimartingale. Indeed if we demand
that JX (f n ) ! JX (f ) with the strongest form of convergence on the domain S- namely
uniform convergence and the weakest form of convergence on the range R0 (⌦, (F⇧ ), P)namely convergence in probability for every t, then it follows that X is a semimartingale.
To be precise, we make the following definition:
D
Definition 5.67. Let X be an r.c.l.l. adapted process. Then X is said to be a weak stochastic
integrator if
f n 2 S, f n ! 0 uniformly ) JX (f n )t ! 0 in probability for each t < 1.
The Bichteler-Dellacherie-Meyer-Mokobodzky Theorem (first proven by Dellacherie
with contributions from Meyer, Mokobodzky and then independently proven by Bichteler)
states that X is a weak stochastic integrator if and only if it is a semimartingale. If X is
a semimartingale then it is a stochastic integrator. Clearly, if X is a stochastic integrator,
then it is a weak stochastic integrator. To complete the circle, we will now show that if X
is a weak stochastic integrator, then it is a semimartingale.
5.8. The Bichteler-Dellacherie-Meyer-Mokobodzky Theorem
199
Towards this goal, we introduce some notation. Let S+ be the class of stochastic
processes f of the form
m
X
fs (!) =
aj+1 (!)1(sj ,sj+1 ] (s)
(5.8.1)
j=0
where 0 = s0 < s1 < s2 < . . . < sm+1 < 1, aj+1 is bounded Fs+j measurable random
variable, 0  j  m, m 1 and let C be the class of stochastic processes f of the form
n
X
fs (!) =
bj+1 (!)1( j (!), j+1 (!)] (s)
(5.8.2)
j=0
T
where 0 = 0  1  2  . . . ,  n+1 < 1 are (F·+ ) bounded stop times and bj+1 is
bounded F +j measurable random variable, 0  j  n, n 1.
For f 2 C, let us define
n
X
IX (f )t (!) =
bj+1 (!)(X j+1 ^t (!) X j ^t (!)).
(5.8.3)
j=0
F
So for f 2 S, JX (f ) = IX (f ). Indeed, if X is a stochastic integrator, then as noted
R
earlier, IX (f ) = f dX, 8f 2 C.
We start with a few observations.
A
Lemma 5.68. Let X be an r.c.l.l. adapted process, f 2 C and ⌧ be a stop time. Then we
have, for all t < 1
IX (f 1[0,⌧ ] )t = IX (f )t^⌧ .
(5.8.4)
R
Proof. Let f be given by (5.8.2) and let g = f 1[0,⌧ ] . Then we have
gs =
n
X
bj+1 1(
D
j=0
and writing dj+1 = bj+1 1{
j ⌧ }
j ^⌧, j+1 ^⌧ ]
(s)
, it follows that dj+1 is F +j ^⌧ measurable and we have
gs =
n
X
dj+1 1(
j=0
j ^⌧, j+1 ^⌧ ]
(s).
Since IX (f ) and IX (g) are defined pathwise, we can verify that the relation (5.8.4) is
true.
Lemma 5.69. Let X be an r.c.l.l. adapted process. Then X is a weak stochastic integrator
if and only if X satisfies the following condition for each t < 1:
8" > 0 9K" < 1 s.t. [
sup
f 2S, |f |1
P(|JX (f )t | > K" )]  ".
(5.8.5)
200
Chapter 5. Semimartingales
Proof. If X satisfies (5.8.5), then given f n 2 S, an = supt,! |ftn (!)| ! 0, we need to show
JX (f n )t ! 0 in probability. So given " > 0, ⌘ > 0, get K" as in (5.8.5). Let n0 be such
that for n n0 , we have an K" < ⌘. Then gn = a1n f n 2 S and |gn |  1. Hence
P(|JX (f n )t |
⌘)  P(|JX (g n )t | > K" )  ".
Thus X is a weak stochastic integrator. Conversely let X be a weak stochastic integrator.
If for some ", t < 1, no such K" < 1 exists, then for each n, we will get f n such that
|f n |  1 and
P(|JX (f n )t | > n)
".
(5.8.6)
Then g n = n1 f n converges uniformly to zero, but in view of (5.8.6),
" 8n
1.
T
P(|JX (g n )t | > 1)
This contradicts the assumption that X is a weak stochastic integrator.
sup
f 2C, |f |1
P(|IX (f )t | > K" )]  ".
(5.8.7)
A
8" > 0 9K" < 1 s.t. [
F
Lemma 5.70. Let X be an r.c.l.l. adapted process. Then X satisfies (5.8.5) if and only if
for each t < 1
Proof. Since f 2 S implies f 1(0,1) 2 C, it is easy to see that (5.8.7) implies (5.8.5).
R
So now suppose K" is such that (5.8.5) holds. We show that (5.8.7) holds. First let
f 2 S+ be given by
m
X1
fs (!) =
aj+1 (!)1(sj ,sj+1 ] (s)
j=0
D
where 0 = s0 < s1 < s2 < . . . < sm < 1, aj+1 is bounded Fs+j measurable random variable,
0  j  (m 1). Let 0 < k < k1 be such that sj + k < sj+1 , 0  j  m and let
gsk (!) =
m
X1
aj+1 (!)1(sj +
k ,sj+1 ]
(s).
j=0
Since Fs+j ✓ Fsj + k , it follows that g k 2 S. Noting that g k converges to f and using the
explicit formula for IX , JX , we see that (the number of terms remain fixed!)
JX (g k )t ! IX (f )t .
Hence we conclude that
P(|IX (f )t | > K" )  ".
5.8. The Bichteler-Dellacherie-Meyer-Mokobodzky Theorem
201
In other words, (5.8.5) implies
8" > 0 9K" < 1 s.t. [
sup
f 2S+ , |f |1
P(|IX (f )t | > K" )]  ".
(5.8.8)
Let us note that if f 2 C is given by (5.8.2) with j being simple stop times, namely taking
finitely many values, then f 2 S+ . To see this, let us order all the values that the stop
times j j = 0, 1, . . . , n take and let the ordered list be s0 < s1 < s2 < s3 < . . . < sk .
Recall that by construction, f is l.c.r.l. adapted and thus
bj = lim f (u)
u#sj
exists and is Fs+j measurable. And then
fs =
k 1
X
bj 1(sj ,sj+1 ] (s).
T
j=0
Now returning to the proof of (5.8.7), let f 2 C be given by
j=0
bj+1 1(
j , j+1 ]
(s)
F
n
X1
fs =
A
with 0 = 0  1  2  . . . ,  n < 1 are (F·+ ) bounded stop times and bj is bounded
F +j measurable random variable, 0  j  (n 1). Let
[2m
j] + 1
,
m 1
2m
be simple stop times decreasing to j , 0  j  n, and let
R
⌧jm =
gsm =
n
X1
m ] (s).
bj+1 1(⌧jm ,⌧j+1
j=0
D
Then g m converges to f and from the explicit expression for IX (g m ) and IX (f ), it follows
that IX (g m ) converges to IX (f ). As noted above g m 2 S+ and hence
P(|IX (g m )t | > K" )  "
and then IX (g m )t ! IX (f )t implies
P(|IX (f )t | > K" )  ".
Since this holds for every f 2 C, (5.8.7) follows.
The preceding two lemmas lead to the following interesting result. The convergence
for each t in the definition of weak stochastic integrator leads to the apparently stronger
result- namely convergence in ducp .
202
Chapter 5. Semimartingales
Theorem 5.71. Let X be a weak stochastic integrator. Then f n 2 C, f n ! 0 uniformly
implies
ucp
IX (f n ) ! 0.
(5.8.9)
As a consequence, X satisfies (4.2.20).
Proof. Fix f n 2 C such that f n ! 0 uniformly. Proceeding as in the proof of Lemma 5.69,
one can show using (5.8.7) that for each t,
IX (f n )t ! 0 in probability.
Fix ⌘ > 0, T < 1. For n
(5.8.10)
1, let
⌧ n = inf{t
0 : |IX (f n )t | > 2⌘}.
T
⌧ n is a stop time with respect to the filtration (F·+ ). Let g n = f n 1[0,⌧ n ] . Note that g n 2 C.
In view of Lemma 5.68 we have
IX (g n )T = IX (f n )T ^⌧ n .
F
Also, from definition of ⌧ n , we have
{ sup |IX (f n )t | > 2⌘} ✓ {|IX (g n )T | > ⌘}.
0tT
(5.8.11)
A
Clearly, g n converges to 0 uniformly and hence as noted in (5.8.10),
IX (g n )t ! 0 in probability.
R
In view of (5.8.11), this proves (5.8.9). As seen in Remark 4.23, (5.8.9) implies (4.2.20).
Here is one last observation in this theme.
D
Theorem 5.72. Let X be a weak stochastic integrator and let hn 2 C. Then
hn
ucp
! 0 ) IX (hn )
Proof. Fix T < 1 and let n1 = 1. For each k
n
ucp
! 0.
(5.8.12)
2, get nk such that nk > nk
1
and
nk ) P( sup |hnt | > k1 )  k1 .
0tT
For nk  n < nk+1 , let
n
= inf{t : |hnt | > k1 }.
Since hn 2 C, (a left continuous step function) it can be seen that
that
f n = hn 1{[0, n ]}
n
is a stop time and
5.8. The Bichteler-Dellacherie-Meyer-Mokobodzky Theorem
203
satisfies
|ftn | 
1
k
for nk  n < nk+1
and
{ sup |hnt | > k1 } = {
0tT
n
 T }.
So
P(
n
 T) 
1
k
for nk  n < nk+1 .
(5.8.13)
Thus
f n converges to 0 uniformly
(5.8.14)
and
In view of Theorem 5.71, (5.8.14) implies IX (f n )
yield the desired conclusion, namely
n
> T ).
(5.8.15)
ucp
! 0 and then (5.8.13) and (5.8.15)
ucp
! 0.
F
IX (hn )
P(
T
P(IX (f n )t = IX (hn )t 8t 2 [0, T ])
A
In view of this result, for a weak stochastic integrator, we can extend IX continuously
to the closure C̄ of C in the ducp metric.
ucp
R
For g 2 C̄, let IX (g) be defined as limit of IX (g n ) where g n ! g. It is easy to see
R
that IX is well defined. If X is a stochastic integrator, IX agrees with gdX.
We identify C̄ in the next result.
D
Lemma 5.73. The closure C̄ of C in the ducp metric is given by
C̄ = {Z : Z 2 R0 }.
ucp
Proof. Let f 2 C̄ and let f n 2 C, f n ! f . Then Vtn = limu#t fun are r.c.l.l. adapted
processes and V n can be seen to be Cauchy in ducp and hence converge to V . Further,
using Theorem 2.69, it follows that a subsequence of V n converges to V uniformly on [0, T ]
for every T < 1, almost surely. Thus V 2 R0 . Now it follows that f = V .
Theorem 5.74. Let X be a weak stochastic integrator and Z 2 R0 . Let {⌧nm : n
a sequence of partitions of [0, 1) via stop times:
0 = ⌧0m < ⌧1m < ⌧2m . . . ; ⌧nm " 1, m
1
1} be
204
Chapter 5. Semimartingales
such that for some sequence
|Zt
for k
# 0 (of real numbers)
m
Z⌧nm | 
m
for ⌧nm < t  ⌧n+1
, n
m
1, let
Ztm
=
1
X
0, m
1
(5.8.16)
m ] (t).
Zt^⌧nm 1(⌧nm ,⌧n+1
n=0
Then
IX (Z m ) =
1
X
n=0
and
m ^t
Zt^⌧nm (X⌧n+1
IX (Z m )
X⌧nm ^t )
ucp
m
# 0, {⌧nm : n
Ztm,k
=
k
X
k
X
A
IX (Z
m,k
)=
n=0
Z m|
m ^t
Zt^⌧nm (X⌧n+1
IX (Z m,k )
ucp
! IX (Z m ).
This proves (5.8.17). Now Z m converges to Z
D
X⌧nm ^t ).
> 0)  P(⌧km < T ) and hence Z m 2 C̄ and
R
Now
P(sup0tT |Ztm,k
1, satisfying (5.8.16) exist, (say
m ] (t).
Zt^⌧nm 1(⌧nm ,⌧n+1
n=0
Then Z m,k 2 C and
1} m
F
Proof. Let us note that given
given in (4.5.12)).
For k 1, let
(5.8.18)
T
! IX (Z ).
R
Thus, if X is also a stochastic integrator, IX (f ) = f dX.
(5.8.17)
IX (Z m )
ucp
uniformly and hence Z 2 C̄ and
! IX (Z ).
If X is also a stochastic integrator, using Theorem 4.57 it follows that for Z 2 R0 ,
R
IX (Z ) agrees with Z dX.
Remark 5.75. Note that while dealing with weak stochastic integrators, we considered the
filtration F·+ for defining the class C, but the definition of weak stochastic integrator did not
require the underlying filtration to be right continuous.
Thus for Z 2 R0 and a weak stochastic integrator X, we define IX (Z ) to be the
R
stochastic integral Z dX. When a weak stochastic integrator is also a stochastic integrator, this does not lead to any ambiguity as noted in the previous Theorem.
5.8. The Bichteler-Dellacherie-Meyer-Mokobodzky Theorem
205
Remark 5.76. A careful look at results in section 4.6 show that Theorem 4.59, Theorem
4.62 continue to be true if the underlying processes are weak stochastic integrators instead of
stochastic integrators. Instead of invoking Theorem 4.57, we can invoke Theorem 5.74. Thus
weak stochastic integrators X, Y admit quadratic variations [X, X], [Y, Y ] and cross-quadratic
variation [X, Y ]. Various results on quadratic variation obtained in section 4.6 continue to be
true for weak stochastic integrators.
Moreover, Theorem 5.37, Theorem 5.38 and Lemma 5.39 are true for weak stochastic
integrators as well since the proof only relies on quadratic variation. Likewise, Lemma 5.41 is
true for weak stochastic integrators since apart from quadratic variation, it relies on (4.2.20),
a property that holds for weak stochastic integrators as noted earlier in Remark 4.23.
T
As a consequence, Theorem 5.43, Theorem 5.48 and Theorem 5.51 are true for weak
stochastic integrators.
F
This discussion leads to the following result, whose proof is same as that of Theorem
5.52.
Theorem 5.77. Let X be a weak stochastic integrator. Then X is a semimartingale.
A
Here is the full version of the Bichteler-Dellacherie-Meyer-Mokobodzky Theorem.
Theorem 5.78. Let X be an r.c.l.l. (F· ) adapted process. Let JX be defined by (4.2.1)(4.2.2). Then the following are equivalent.
R
(i) X is a weak stochastic integrator, i.e. if f n 2 S, f n ! 0 uniformly, then JX (f n )t ! 0
in probability 8t < 1.
D
(ii) If f n 2 S, f n ! 0 uniformly, then JX (f n )
(iii) If f n 2 S, f n
ucp
! 0 , then JX (f n )
bp
(iv) If f n 2 S, f n ! 0 , then JX (f n )
ucp
! 0.
ucp
! 0.
ucp
! 0.
(v) X is a stochastic integrator, i.e. the mapping JX from S to R0 (⌦, (F⇧ ), P) has an
e P) 7! R0 (⌦, (F⇧ ), P) satisfying
extension JX : B(⌦,
bp
f n ! f implies JX (f n )
ucp
! JX (f ).
(vi) X is a semimartingale, i.e. X admits a decomposition X = M + A where M is a local
martingale and A is a process with finite variation paths.
206
Chapter 5. Semimartingales
(vii) X admits a decomposition X = N + B where N is a locally square integrable martingale and B is a process with finite variation paths.
Proof. Equivalence of (v), (vi), (vii) has been proven in Theorem 5.52 and Corollary 5.53.
Clearly,
(v) ) (iv) ) (iii) ) (ii) ) (i).
Theorem 5.71 and Theorem 5.72 tell us that
(i) ) (ii) ) (iii).
And we have observed in Theorem 5.77 that (iii) implies (vi). This completes the proof.
Enlargement of Filtration
T
5.9
A
F
The main result of the section is about enlargement of the underlying filtration (F⇧ ) by
adding a set A 2 F to each Ft . The surprising result is that a semimartingale for the
original filtration remains a semimartingale for the enlarged filtration. In the traditional
approach, this was a deep result as it required decomposition of the semimartingale into
local martinagle w.r.t. the enlarged filtration and a finite variation process.
Let A 2 F be fixed and let us define a filtration (G⇧ ) by
Gt = {(B \ A) [ (C \ Ac ) : B, C 2 Ft }.
(5.9.1)
D
R
It is easy to see that Gt is a -field for all t
0 and (G⇧ ) is a filtration. Using the
description (5.9.1) of sets in Gt , it can be seen that if ⇠ is a Gt measurable bounded random
variable, then 9 Ft measurable bounded random variables ⌘, ⇣ such that
⇠ = ⌘ 1A + ⇣ 1Ac .
(5.9.2)
Let S(G⇧ ) denote the class of simple predictable process for the filtration (G⇧ ). Using (5.9.2)
it is easy to verify that for f 2 S(G⇧ ) 9 g, h 2 S(F⇧ ) such that
e
f (t, !) = g(t, !)1A (!) + h(t, !)1Ac (!) 8(t, !) 2 ⌦.
(5.9.3)
With this we can now describe the connection between predictable processes for the two
filtrations.
Theorem 5.79. Let f be an (G⇧ )-predictable process. We can choose (F⇧ ) predictable
processes g, h such that (5.9.3) holds. Further, if f is bounded by a constant c, then g, h
can also be chosen to be bounded by the same constant c.
5.9. Enlargement of Filtration
207
Proof. Let K be the set of (G⇧ )-predictable process f for which the conclusion is true.
We have seen that K contains S(G⇧ ). We next show that K is closed under pointwise
convergence. This will show in particular that K is closed under bp-covergence and hence
that the conclusion is true for all bounded predictable processes. If f n 2 K with
f n = g n 1 A + hn 1 A c
and f n ! f then we can take
gt (!) = lim sup gtn (!)1{lim supn!1 |gtn |(!)<1}
(5.9.4)
ht (!) = lim sup hnt (!)1{lim supn!1 |hnt |(!)<1}
(5.9.5)
n!1
n!1
T
and then it follows that f = g 1A + h1Ac and thus f 2 K. As noted above, this proves
K contains all bounded predictable process and in turn all predictable processes since
f n = f 1{|f n |n} converges pointwise to f .
If f is bounded by c, we can replace g by g̃ = g 1{|g|c} and h by h̃ = h1{|h|c} .
A
F
Theorem 5.80. Let X be a semimartingale for the filtration (F⇧ ). Let A 2 F and (G⇧ ) be
defined by (5.9.1).
Then X is also a semimartingale for the filtration (G⇧ ).
R
Proof. Let us denote the mapping defined by (4.2.1) and (4.2.2) for the filtration (G⇧ ) by
HX and let JX be the mapping for the filtration (F⇧ ). Since these mappings are defined
pathwise, it is easy to verify that if f 2 S(G⇧ ) and g, h 2 S(F⇧ ) are as in (5.9.3), then
HX (f ) = JX (g)1A + JX (h)1Ac
(5.9.6)
D
We will prove that X is a weak-stochastic integrator for the filtration (G⇧ ). For this, let
e P(G⇧ )) decrease to 0 uniformly. Let an # 0 be such that |f n |  an . Then for each
f n 2 B(⌦,
n invoking Theorem 5.79 we choose g n , hn 2 S(F⇧ ) with |g n |  an , |hn |  an such that
f n = g n 1A + g n 1Ac .
As noted above this gives, for n
1
HX (f n ) = JX (g n )1A + JX (hn )1Ac .
(5.9.7)
Since X is a semimartingale for the filtration (F⇧ ), it is a stochastic integrator. Thus,
ucp
ucp
ucp
JX (g n ) ! 0 and JX (hn ) ! 0 and then (5.9.7) implies HX (f n ) ! 0. Hence X is a
weak-stochastic integrator for the filtration (F⇧ ). Invoking Theorem 5.78, we conclude that
X is a semimartingale for the filtration (G⇧ ).
208
Chapter 5. Semimartingales
Remark 5.81. As noted in Theorem 4.11, when f is (F⇧ )-predictable bounded process,
R
HX (f ) = JX (f ) and thus the stochastic integral f dX is unambiguously defined.
Theorem 5.82. Let X be a stochastic integrator on (⌦, F, P) with a filtration (F⇧ ). Let
Q be a probability measure on (⌦, F) that is absolutely continuous w.r.t. P. Then X is a
stochastic integrator on (⌦, F, Q) and the stochastic integral under P is a version of the
integral under Q.
Proof. Let H be the completion of F under Q and let Ht be the -field obtained by adding
all Q-null sets in every Ht . It can be checked that a process f is (H⇧ )-predictable if and
only if f 1⌦0 is (G⇧ )-predictable. Let
e P(H⇧ )).
HX (f ) = JX (f 1⌦0 ) f 2 B(⌦,
T
It is easy to see that HX is the required extension of the integral of simple predictable
processes.
A
F
Remark 5.83. Suppose we start with a filtration (F̃⇧ ) that may not satisfy the condition that
each F̃t contains all null sets. Suppose X is a (F̃⇧ )-adapted process that satisfies (4.10.2) for
this filtration. Let Ft be the smallest -field containing F̃t and all the null sets. It is easy
to see that X continues to satisfy (4.10.2) w.r.t. the filtration (F⇧ ) and is thus a stochastic
integrator.
R
Exercise 5.84. Let X be a semimartingale for a filtration (F⇧ ) on (⌦, F, P). Let {Am :
m 1} be a partition of ⌦ with An 2 F for all n 1. For t 0 let
Gt = (Ft [ {Am : m
Show that
1}).
D
(i) For every (G⇧ )-predictable process f , there exists (F⇧ )-predictable processes f m such that
1
X
f=
1Am f m .
m=1
(ii) Suppose for each m
as n ! 1. Let
1, {Y m,n : n
Zn =
1} are r.c.l.l. processes such that Y m,n
1
X
1Am Y m,n
m=1
and
Z=
1
X
m=1
1Am Y m .
ucp
! Ym
5.9. Enlargement of Filtration
Then prove that Z n
209
ucp
! Z as n ! 1.
D
R
A
F
T
(iii) Show that X is a stochastic integrator for the filtration (G⇧ ).
Chapter 6
Pathwise Formula for the Stochastic Integral
Preliminaries
T
6.1
R
A
F
R
For a simple predictable process f , the stochastic integral f dX has been defined explicitly,
Rt
path by path. In other words, the path t 7! ( 0 f dX)(!) is a function of the paths
{fs (!) : 0  s  t} and {Xs (!) : 0  s  t} of the integrand f and integrator X. For
a general (bounded) predictable f the integral has been defined as limit in probability of
suitable approximations and it is not clear if we can obtain a pathwise version. In statistical
inference for stochastic process the estimate, in stochastic filtering theory the filter and in
stochastic control theory the control in most situations involves stochastic integral, where
the integrand and integrator are functionals of the observation path and to be meaningful,
the integral should also be a functional of the observation.
D
How much does the integral depend upon the underlying filtration or the underlying
probability measure? Can we get one fixed version of the integral when we have not one
but a family of probability measures {P↵ } such that X is a semimartingale under each P↵ .
If we have one probability measure Q such that each P↵ is absolutely continuous w.r.t.
Q and the underlying process X is a semimartingale under Q then the answer to the
question above is yes- simply take the integral defined under Q and that will agree with
the integral under P↵ for each ↵ by Remark 4.15.
When the family {P↵ } is countable family, such a Q can always be constructed. However, such a Q may not exist in general. Once concrete situation where such a situation
arises and has been considered in the literature is when we have a Markov Process- when
one considers the family of measures {Px : x 2 E}, where Px represents the distribution of
Markov Process conditioned on X0 = x. See [18]. In this context, Cinlar, Jacod, Protter
210
6.2. Pathwise Formula for the Stochastic Integral
211
and Sharp [9] showed the following : For processes S such that S is semimartingale under
Px for every x and for f in a suitable class of predictable processes, there exists a process
R
Z such that Z is a version of f dS under Px for every x.
In Bichteler [3], Karandikar [32], [33], [37] it was shown that for an r.c.l.l. adapted
process Z and a semimartingale X, suitably constructed Riemann sums converge almost
R
surely to the stochastic integral Z dX . This result was recast in [40] to obtain a universal
mapping : D([0, 1), R) ⇥ D([0, 1), R) 7! D([0, 1), R) such that if X is a semimartingale
and Z is an r.c.l.l. adapted process, then (Z, X) is a version of the stochastic integral
R
Z dX.
As in the previous chapter, we fix a filtration (F⇧ ) on a complete probability space
(⌦, F, P) and we assume that F0 contains all P-null sets in F.
T
First we will prove a simple result which enables us to go from L2 estimates to almost
sure convergence.
1
X
F
Lemma 6.1. Let V m be a sequence of r.c.l.l. process and ⌧k an increasing sequence of stop
times, increasing to 1 such that
ksup |Vtm |k2 < 1.
(6.1.1)
A
m=1 t⌧k
Then we have
sup|Vtm | ! 0 8T < 1, a.s.
R
tT
Proof. The condition (6.1.1) implies
D
k
and hence for each k,
1
X
sup |Vtm |k2 < 1
m=1 t⌧k
1
X
sup |Vtm | < 1 a.s.
m=1 t⌧k
Since ⌧k increase to 1, the required result follows.
6.2
Pathwise Formula for the Stochastic Integral
Recall that D([0, 1), R) denotes the space of r.c.l.l. functions on [0, 1) and that for ↵ 2
D([0, 1), R), ↵(t ) denotes the left limit at t (for t > 0) and ↵(0 ) = 0.
212
Chapter 6. Pathwise Formula for the Stochastic Integral
Fix ↵ 2 D([0, 1), R). For each n
1; let {tni (↵) : i
follows : tn0 (↵) = 0 and having defined tni (↵), let
tni+1 (↵) = inf{t > tni (↵) : |↵(t)
↵(tni (↵))|
n
2
1} be defined inductively as
↵(tni (↵))|
or |↵(t )
2
n
}. (6.2.1)
Note that for each ↵ 2 D([0, 1), R) and for n 1, tni (↵) " 1 as i " 1 (if limi tni (↵) = t⇤ <
1, then the function ↵ cannot have a left limit at t⇤ ). For ↵, 2 D([0, 1), R) let
n (↵,
)(t) =
1
X
i=0
↵(tni (↵) ^ t)( (tni+1 (↵) ^ t)
(tni (↵) ^ t)).
(6.2.2)
D⇤ = {(↵, ) :
n (↵,
2 D([0, 1), R)
n (↵,
n
:0
if (↵, ) 2 D⇤
)
(6.2.3)
otherwise.
A
(↵, ) =
8
<lim
) converges in ucc topology}
F
and for ↵,
T
Since tni (↵) increases to infinity, for each ↵ and t fixed, the infinite sum appearing above
is essentially a finite sum and hence n (↵, ) is itself an r.c.l.l. function. We now define a
mapping : D([0, 1), R) ⇥ D([0, 1), R) 7! D([0, 1), R) as follows: Let D⇤ ✓ D([0, 1), R) ⇥
D([0, 1), R) be defined by
Note that the mapping has been defined without any reference to a probability measure
or a process. Here is the main result on Pathwise integration formula.
R
Theorem 6.2. Let X be a semimartingale on a probability space (⌦, F, P) with filtration
(F⇧ ) and let U be an r.c.l.l. adapted process. Let
Then
D
Z· (!) =
Z=
n
i
Proof. For each fixed n, define {
n
i+1
= inf{t >
n
i
: i
: |Ut
(U· (!), X· (!))
(6.2.4)
Z
(6.2.5)
U dX.
0} inductively with
U in |
2
n
or |Ut
and is thus a stop time (for all n, i). Let us note that
n (U· (!), X· (!)). Then we can see that
Ztn =
1
X
j=0
Ut^ jn (Xt^
n
j+1
n
i (!)
Xt^ jn )
n
0
= 0 and
U in |
2
n
}
= tni (U· (!)). Let Z·n (!) =
6.2. Pathwise Formula for the Stochastic Integral
and thus Z n =
R
213
U n dX where
Utn =
1
X
Ut^ jn 1(
n n
j , j+1 ]
(t).
j=0
Since by definition of {
n
i },
{U n }, we have
|Utn
Un
Ut |  2
n
R
Zn
(6.2.6)
and hence
! U in ucp. Then by Theorem 4.46,
! Z = U dX in the ucp
metric.
The crux of the argument is to show that the convergence is indeed almost sureZ t
Z t
n
sup| U dX
U dX| ! 0 8T < 1 a.s.
(6.2.7)
tT
0
0
F
T
Once this is shown, it would follow that (U· (!), X· (!)) 2 D⇤ a.s. and then by definition of
and Z we conclude that Zn = (U n , X) converges to (U, X) in ucc topology almost
surely. Since Z n ! Z in ucp, we have Z = (U, X) completing the proof.
Remains to prove (6.2.7). For this, first using Corollary 5.53, let us decompose X as
X = M + A, M 2 M2loc and A 2 V. Now using the fact that the dA integral is just the
Lebesgue - Stieltjes integral and the estimate (6.2.6) we get
Z t
Z t
Z t
n
| U dA
U dA| 
|Usn Us |d|A|t
0
A
0
0
2
and hence
Z
t
Z
U n dA
R
sup|
tT
0
n
|A|t .
t
0
U dA|  2
n
|A|T
(6.2.8)
! 0.
D
Thus (6.2.7) would follow in view of linearity of the integral once we show
Z t
Z t
n
[sup| U dM
U dM | ] ! 0 8T < 1 a.s.
tT
0
(6.2.9)
0
Let ⌧k be stop times increasing to 1 such that ⌧k  k and M [⌧
martingale so that
k
k
E[ [M, M ]⌧k ] = E[ [M [⌧ ] , M [⌧ ] ]k ] < 1.
k]
is a square integrable
(6.2.10)
Thus using the estimate (5.4.10) on the growth of the stochastic integral with respect to a
local martingale (Theorem 5.26), we get
Z t
Z t
Z ⌧k
E[sup | U n dM
U dM |2 ]  4E[
|Usn Us |2 d[M, M ]s ]
t⌧k 0
0
0
(6.2.11)
 4(2
2n
)E[ [M, M ]⌧k ].
214
Chapter 6. Pathwise Formula for the Stochastic Integral
Thus, writing ⇠tn =
Rt
0
U n dM
Rt
0
U dM and ↵k =
ksup |⇠tn |k2  2
p
E[ [M, M ]⌧k ], we have
n+1
t⌧k
↵k .
(6.2.12)
Since ↵k < 1 as seen in (6.2.10), Lemma 6.1 implies that (6.2.9) is true completing the
proof.
T
R
Remark 6.3. This result implies that the integral U dX for an r.c.l.l. adapted process U
does not depend upon the underlying filtration or the probability measure or on the decomposition of the semimartingale X into a (local) martingale and a process with finite variation
R·
paths. A path ! 7! 0 U dX(!) of the integral depends only on the paths ! 7! U· (!) of
the integrand and ! 7! X· (!) of the integrator. The same however cannot be said in general
R
about f dX if f is given to be a predictable process.
Remark 6.4. In Karandikar [37, 40] the same result was obtained with tni (↵) defined via
tni (↵) : |↵(t)
↵(tni (↵))|
2
n
}
F
tni+1 (↵) = inf{t
Pathwise Formula for Quadratic Variation
R
6.3
A
instead of (6.2.1). The result is of course true, but requires the underlying -fields were required
to be right continuous and to prove that the resulting jn are stop times required deeper results
which we have avoided in this book.
D
In Section 5.2 we had obtained a pathwise formula for the quadratic variation process of
a locally square integrable martingale - namely (5.3.16). We now observe that the same
formula gives quadratic variation of local martingales as well - indeed of semimartingales.
We will need to use notations from Section 5.2 as well as Section 6.2
Theorem 6.5. Let
gale X, let
be the mapping defined in Section 5.2 by (5.2.2). For a semimartin[X, X]t (!) = [ (X.(!))](t).
(6.3.1)
Then [X, X] is a version of the quadratic variation [X, X] i.e.
P( [X, X]t = [X, X]t 8t) = 1.
Proof. For ↵ 2 D([0, 1), R) and for n
1, let {tni (↵) : i
1} be defined inductively
by ba8. Recall the definition (5.2.1) of n (↵) and (6.2.2) of n . Using the identity
6.3. Pathwise Formula for Quadratic Variation
215
(b a)2 = b2 a2 2a(b a) with b = ↵(tni+1 (↵) ^ t) and a = ↵(tni (↵) ^ t) and summing
over i 2 {0, 1, 2 . . .}, we get the identity
n (↵)
Let
= (↵(t))2
e be as defined in Section 5.2 and
,D
(↵(0))2
2
n (↵).
(6.3.2)
, D⇤ be as defined in Section 6.2. Let
b = {↵ 2 D : (↵, ↵) 2 D⇤ }.
D
Then using (6.3.2) along with the definition (6.2.3) of
and
(↵) = (↵(t))2
b✓D
e
D
, it follows that
b
2 (↵) ↵ 2 D.
(↵(0))2
T
As noted in Section 6.2, (X· (!), X· (!)) 2 D⇤ and
Z
X dX = (X, X)
(6.3.3)
(6.3.4)
X02
A
[X, X]t =
Xt2
F
From aiy40s, (6.3.3) and (6.3.4) and it follows that
2
Z
t
X dX.
0
D
R
This along with (4.6.1) implies [X, X]t = [X, X]t completing the proof.
(6.3.5)
Chapter 7
Continuous Semimartingales
D
R
A
F
T
In this chapter we will consider continuous semimartingales and show that stochastic differential equations driven by these can be analyzed essentially using the same techniques
as in the case of SDE driven by Brownian motion. This can be done using random time
change. The use of random time change in study of solutions to stochastic di↵erential
equations was introduced in [32], [33].
R
We introduce random time change and we then obtain a growth estimate on f dX
where X is a continuous semimartingale and f is a predictable process. Then we observe
R
that if a semimartingale satisfies a condition (7.2.2), then the growth estimate on f dX
R
is very similar to the growth estimate on f d , where
is a brownian motion. We
also note that by changing time via a suitable random time, any semimartingale can be
transformed to a semimartingale satisfying (7.2.2). Thus, without loss of generality we
can assume that the driving semimartingale satisfies (7.2.2) and then use techniques used
for Brownian motion case. We thus show that stochastic di↵erential equation driven by
continuous semimartingales admit a solution when the coefficients are Lipschitz functions.
We also show that in this case, one can get a pathwise formula for the solution, like the
formula for the integral obtained in the previous chapter.
7.1
Random Time Change
Change of variable plays an important role in calculations involving integrals of functions
of a real variable. As an example, let G be a continuous increasing function with G[0] = 0.
Let us look at the formula
Z
t
f (G(t)) = f (0) +
0
216
f 0 (G(s))dG(s).
(7.1.1)
7.1. Random Time Change
217
which was derived in section 4.8. We had seen that when G is absolutely continuous, this
formula follows by the chain rule for derivatives. Let a : [0, 1) 7! [0, 1) be a continuous
strictly increasing one-one onto function. Let us write G̃(s) = G(a(s)). It can be seen that
(7.1.1) can equivalently be written as
Z t
f (G̃(t)) = f (0) +
f 0 (G̃(s))dG̃(s).
(7.1.2)
0
Exercise 7.1. Show that (7.1.1) holds if and only if (7.1.2) is true.
So to prove (7.1.1), suffices to prove (7.1.2) for a suitable choice of a(t). Let
a(s) = inf{t
0 : (t + G(t))
s}.
F
T
For this choice of a it can be seen that G̃ is a continuous increasing function and that for
0  u  v < 1, G̃(v) G̃(u)  v u so that G̃ is absolutely continuous and thus (7.1.2)
follows from chain rule.
When working with continuous semimartingales, the same idea yields interesting resultsof course, the time change t 7! a(t) has to be replaced by t 7! t , where t is a stop time.
A
Definition 7.2. A (F⇧ )-random time change = ( t ) is a family of (F⇧ )- stop times { t :
0  t < 1} such that for all ! 2 ⌦, t 7! t (!) is a continuos strictly increasing function from
[0, 1) onto [0, 1).
Example 7.3. Let A be a (F⇧ )-adapted continuous increasing process with A0 = 0. Then
s
= inf{t
0 : (t + At )
s}
R
can be seen to be a (F⇧ )-random time change.
D
Example 7.4. Let B be a (F⇧ )-adapted continuous increasing process with B0 = 0 such that
B is strictly increasing and limt!1 Bt = 1 a.s.. Then
s
= inf{t
0 : Bt
s}
can be seen to be a (F⇧ )-random time change.
Recall the Definition 2.35 of the stopped -field. Given a (F⇧ )-random time change
= ( t ), we define a new filtration (G⇧ ) = (Gt ) as follows:
Gt = F t , 0  t < 1.
(7.1.3)
Clearly, for s  t, we have s  t and hence Gs ✓ Gt and so {Gs } is a filtration. Further,
G0 = F0 . We will denote the filtration (G⇧ ) defined by (7.1.3) as ( F⇧ ). Given a process f ,
we define the process g = [f ] via
gs = f
s
0  s < 1.
(7.1.4)
218
Chapter 7. Continuous Semimartingales
The map f ! g is linear and preserves limits: if f n converges to f pointwise or in ucp then
so does the sequence g n . We also define = { t : 0  t < 1} via
t
and denote
by [ ]
1.
Here
Exercise 7.5. Show that if
= inf{s
0 :
u (!)
= v then
v (!)
= u.
= [⌧ ] by
= [ ]⌧ 1 =
1
(7.1.5)
is the reverse time change.
Given a (F⇧ )- stop time ⌧ , we define
Note the appearance of [ ]
t}
s
⌧.
(7.1.6)
in the definition above. It is not difficult to see that
T
= ⌧.
Recall the definition (4.4.12) of X [⌧ ] , the process X stopped at ⌧ . For Y = [X], note that
[⌧ ]
s
=X
s ^⌧
=X
and thus we have.
Y[
]
=X
s^
s^
= Ys^ = Ys[
]
F
( [X [⌧ ] ])s = X
= X [⌧ ] .
(7.1.7)
A
We will now prove few relations about random time change and its interplay with
notions discussed in the earlier chapters such as stop times, predictable processes, local
martingales, semimartingales and stochastic integrals.
=(
s)
= [ ]
1
be defined via
is a (G⇧ )- random time change.
D
(i)
R
Theorem 7.6.
= ( t ) be a (F⇧ )- random time change. Let
(7.1.5). Then we have
(ii) Let ⌧ be a (F⇧ )-stop time. Then
= [⌧ ] =
⌧
is a (G⇧ )-stop time.
(iii) Let U be a (F⇧ ) adapted r.c.l.l. process. Then V =
process.
[U ] is a (G⇧ ) adapted r.c.l.l.
(iv) Let f be a (F⇧ ) bounded predictable process. Then g = [f ] is a (G⇧ ) bounded predictable process. If f is a (F⇧ ) locally bounded predictable process then g = [f ] is a
(G⇧ ) locally bounded predictable process.
(v) Let A be a (F⇧ ) adapted r.c.l.l. process with finite variation paths. Then B = [A] is
a (G⇧ ) adapted r.c.l.l. process with finite variation paths.
7.1. Random Time Change
219
(vi) Let M be a (F⇧ ) local martingale. Then N = [M ] is a (G⇧ ) local martingale.
(vii) Let X be a (F⇧ ) semimartingale. Then Y = [X] is a (G⇧ ) semimartingale.
(viii) Let Z =
R
f dX. Then [Z] =
R
gdY (where f, g, X, Y are as in (iv) and (vii) ).
(ix) [Y, Y ] = [ [X, X] ], where X, Y are as in (vii).
(x) Let X n , X be (F⇧ ) adapted r.c.l.l. processes such that X n
ucp
[X n ] ! Y = [X].
! X. Then Y n =
is (G⇧ ) adapted. For any a, t 2 [0, 1), note that
Proof. Note that by Corollary 2.50,
{
ucp
a
 t} = {a 
t }.
F
T
Since t is F t = Gt measurable, it follows that { a  t} 2 Gt and hence a is a (G⇧ ) - stop
time. Since s 7! s is continuous strictly increasing function from [0, 1) onto itself, same
is true of s 7! s and hence = ( s ) is a random time change. This proves (i).
Now, for s 2 [0, 1)
{  s} = {⌧ 
s}
2F
s
= Gs .
D
R
A
Thus is a (G⇧ ) - stop time.
For (iii) since X is r.c.l.l., (F⇧ ) adapted and s is a (F⇧ ) - stop time, using Lemma 2.36,
we conclude that Ys = X s is Gs = F s measurable. Thus Y is (G⇧ ) adapted and is clearly
r.c.l.l.
For (iv) the class of bounded processes f such that g = [f ] is (G⇧ )-predictable is
bp-closed and by the part proved already, it contains bounded continuous (F⇧ ) adapted
processes and thus also contains bounded (F⇧ )-predictable processes. Now if f is (F⇧ )predictable and locally bounded, let ⌧ n be sequence of stop times, ⌧ n " 1 such that
n
fn = f [⌧ ] is bounded predictable. Then as shown above, [fn ] is also bounded predictable.
Let n = [⌧ n ]. As seen in (7.1.7),
g[
n]
n
= [f [⌧ ] ] = [fn ]
n
and thus g [ ] is predictable. Now ⌧ n " 1 implies n " 1 and thus g is locally bounded
(G⇧ )-predictable process. This proves (iv).
For (v), we have already noted that B is (G⇧ ) adapted. And clearly,
Var[0,s] (B(!)) = Var[0, s (!)] (A(!))
and hence paths of B have finite variation.
220
n
Chapter 7. Continuous Semimartingales
For (vi), in order to prove that N is a (G⇧ ) local martingale, we will obtain a sequence
of (G⇧ ) stop times increasing to 1 such that for all (G⇧ )- stop times ⇣,
E[N
n ^⇣
] = E[N0 ].
This will prove N [ n ] is a martingale and hence that N is a local martingale. Since M is a
local martingale, let ⌧˜n be stop times such M [⌧˜n ] is a martingale for each n where ⌧˜n " 1.
Let
⌧n = ⌧˜n ^ n ^ n .
Then ⌧n  ⌧˜n and ⌧n " 1. Then M [⌧n ] is a martingale for each n and hence for all stop
times ⌘ one has
E[M⌧n ^⌘ ] = E[M0 ].
(7.1.8)
⇣.
n ^⇣
it follows that
n

n
= n. Now for any
] = E[N0 ].
(7.1.9)
Then by part (ii) above, ⌘ is a (F⇧ )-stop time. Note that
N
n ^⇣
F
Let ⌘ = [⇣] =
n,
T
Now let n = [⌧n ] = ⌧n . Since ⌧n 
(G⇧ )-stop time ⇣, we will show
E[N
= M⌧n ^⌘ .
(7.1.10)
D
R
A
Further, M0 = N0 and thus (7.1.8) and (7.1.10) together imply (7.1.9) proving (vi).
Part (vii) follows from (v) and (vi) by decomposing the semimartingale X into a local
martingale M and a process with finite variation paths A : X = M + A. Then [X] =
Y = [M ] + [A] = N + B.
We can verify the validity of (viii) when f is a simple predictable process and then
easy to see that the class of processes for which (viii) is true is bp-closed and thus contains
all bounded predictable processes. We can then get the general case (of f being locally
bounded) by localization.
For (ix), note that
Z t
2
2
[X, X] = Xt X0 2
X dX.
0
Changing time in this equation, and using (viii), we get
Z t
2
2
[[X, X]]t =X t X0 2
X dX
0
Z t
2
2
=Yt
Y0 2
Y dY
0
=[Y, Y ].
7.2. Growth estimate
221
For the last part, note that for T < 1, T0 < 1,
P( sup |Ysn
0tT
)  P( sup |Xsn
Ys |
0tT0
> 0 one has (using Ys = X s )
Xs |
) + P(
T
T0 )
Now given T < 1, > 0 and ✏ > 0, first get T0 such that P( T T0 ) < 2✏ and then for this
ucp
T0 , using X n ! X get n0 such that for n n0 one has P(sup0tT0 |Xsn Xs |
) < 2✏ .
Now, for n n0 we have
P( sup |Ysn
0tT
Ys |
) < ✏.
Remark 2.67 now completes the proof.
T
Remark 7.7. It should be noted that if M is a martingale then N = [M ] may not be a
(G⇧ )-martingale. In fact, Nt may not be integrable as seen in the next exercise.
(ii)
t
is a stop time for each t.
= ( t ) is a random time change.
A
(i)
F
Exercise 7.8. Let W be a Brownian motion and ⇠ be a (0, 1)-valued random variable
independent of W such that E[⇠] = 1. Let Ft = (⇠, Ws : 0  s  t). Let t = t⇠. Show
that
(iii) E[|Zt |] = 1 for all t > 0 where Z = [W ].
Growth estimate
D
7.2
R
(iv) Z is a local martingale but not a martingale.
Let X be a continuous semimartingale and let X = M + A be the decomposition of X with
M being a continuous local martingale, A being a process with finite variation paths. We
will call this as the canonical decomposition. Recall that the quadratic variation [M, M ] is
itself a continuous process and |A|t = Var[0,t] (A) is also a continuous process. For a locally
bounded predictable process f , for any stop time such that the right hand side in (7.2.1)
below is finite one has
Z s
E[ sup |
f dX|2 ]
0s
0+
(7.2.1)
Z
Z
2
2
 8E[
|fs | d[M, M ]s ] + 2E[(
|fs |d|A|s ) ].
0+
0+
222
Chapter 7. Continuous Semimartingales
R
R
R
To see this, we note f dX = f dM + f dA and for the dM integral we use Theorem
R
R
5.26 and for the dA integral we use | f dA|  |f |d|A|.
For process A, B 2 V+ (increasing adapted processes), we define A << B if Ct = Bt At
is an increasing process. The following observation will be used repeatedly in the rest of
this chapter: if A << B, then for all f
Z t
Z t
|
fs dAs | 
|fs |dBs .
0+
0+
We introduce a notion of a amenable semimartingale and obtain a growth estimate on
integrals w.r.t. a amenable semimartingale which is similar to the one for Brownian motion.
[N, N ]s  (t
[N, N ]t
s), |B|t
T
Definition 7.9. A continuous semimartingale Y is said to be a amenable semimartingale if
the canonical decomposition Y = N + B satisfies, for 0  s  t < 1
|B|s  (t
s).
(7.2.2)
0+
0+
A
0s ^T
F
Theorem 7.10. Suppose Y is a continuous amenable semimartingale. Then for any locally
bounded predictable f , and a stop time , one has
Z s
Z ^T
E[ sup |
f dX|2 ]  2(4 + T )E[
|fs |2 ds].
(7.2.3)
R
Proof. The condition (7.2.2) implies that t
This observation along with (7.2.1) yields
Z s
Z
2
E[ sup |
f dX| ]  8E[
0s ^T
0+
[N, N ]t and t
^T
0+
2
|B|t are increasing processes.
|fs | ds] + 2(E[
Z
^T
0+
|fs |ds])2 .
D
Now the required estimate follows by the Cauchy-Schwarz inequality:
Z ^T
Z ^T
2
(E[
|fs |ds])  T E[
|fs |2 ds].
0+
0+
Remark 7.11. The condition (7.2.2) can be equivalently stated as
s
[N, N ]s  t
[N, N ]t , s
|B|s  t
|B|t for 0  s  t < 1
(7.2.4)
or, writing It = t, it is same as
[N, N ] << I,
|B| << I.
(7.2.5)
R
Remark 7.12. We see that for a amenable semimartingale X, the stochastic integral f dX
satisfies a growth estimate similar to the one when X is a Brownian motion. Thus, results such
7.2. Growth estimate
223
as existence, uniqueness, approximation of solution to an SDE driven by a Brownian motion
continue to hold for a amenable semimartingale X. We will come back to this later in this
chapter.
Remark 7.13. In the definition of amenable semimartingale, instead of (7.2.2) we could have
required that for some constant K < 1
[N, N ]t
[N, N ]s  K(t
s), |B|t
|B|s  K(t
s).
(7.2.6)
The only di↵erence is that a constant K 2 would appear in the estimate (7.2.3)
Z ^T
Z s
E[ sup |
f dX|2 ]  2(4 + T )K 2 E[
|fs |2 ds].
0s ^T
0+
(7.2.7)
0+
T
A simple but important observation is that given a continuous semimartingale X one
can get a random time change = ( · ) such that the semimartingale Y = [X] satisfies
(7.2.2). Indeed, given finitely many semimartingales, we can choose one random time
change that does it as we see in the next result.
A
F
Theorem 7.14. Let X 1 , X 2 , . . . , X m be continuous semimartingales, 1  j  m with
respect to the filtration (F⇧ ). Then there exists a random time change = ( t ) (with respect
to the filtration (F⇧ )) such that for 1  j  m, Y j = [X j ] is a amenable semimartingale.
R
Proof. For 1  j  m, let X j = M j + Aj be the canonical decomposition of the semimartingale X j with M j being a continuous local martingale, M0j = 0 and Aj are continuous
processes with finite variation paths. Define an increasing process V by
m
X
Vt = t +
([M j , M j ]t + |Aj |t ).
j=1
D
Then V is strictly increasing adapted process with V0 = 0. Now defining
t
= inf{s
0 : Vs
t}
it follows that
= ( t ) is a random time change. As noted earlier, Y j = [X j ] is a
semimartingale with canonical decomposition Y j = N j + B j where N j = [M j ], B j =
[Aj ]. Further, observing that for 1  j  m, 0  s  t < 1,
[N j , N j ]t
[N j , N j ]s = [M j , M j ]
t
[M j , M j ]
s
V
t
V
s
and
|B j |t
|B j |s = |Aj |
t
|Aj |
s
V
t
V
s
=t
s,
it follows that this random time change satisfies the required condition.
=t
s
224
7.3
Chapter 7. Continuous Semimartingales
Stochastic Di↵erential Equations
Let us consider the Stochastic Di↵erential Equation (3.5.1) where instead of a Brownian
motion as in Chapter 3, here W = (W 1 , W 2 , . . . , W d ) is a amenable semimartingale. The
growth estimate (7.2.3) enables one to conclude that in this case too, Theorem 3.26 is
true and the same proof works essentially- using (7.2.3) instead of (3.4.4). Moreover,
using random time change, one can conclude that the same is true even when W is any
continuous semimartingale. We will prove this along with some results on approximations
to the solution of an SDE.
T
We are going to consider the following general framework for the SDE driven by continuous semimartingales, where the evolution from a time t0 onwards could depend upon
the entire past history of the solution rather than only on its current value as was the case
in equation (3.5.1) driven by a Brownian motion.
Let Y 1 , Y 2 , . . . Y m be continuous semimartingales w.r.t. the filtration (F⇧ ). Here we
will consider an SDE
F
dUt = b(t, ·, U )dYt , t
0, U0 = ⇠0
(7.3.1)
a : [0, 1) ⇥ ⌦ ⇥ Cd ! L(d, m)
(7.3.2)
(t, !) 7! a(t, !, ⇣) is an r.c.l.l. (F⇧ ) adapted process
(7.3.3)
R
be such that for all ⇣ 2 Cd ,
A
where the functional b is given as follows. Recall that Cd = C([0, 1), Rd ). Let
and there is an increasing r.c.l.l. adapted process K such that for all ⇣1 , ⇣2 2 Cd ,
sup ka(s, !, ⇣2 )
D
0st
a(s, !, ⇣1 )k  Kt (!) sup |⇣2 (s)
0st
⇣1 (s)|
(7.3.4)
and b : [0, 1) ⇥ ⌦ ⇥ Cd ! L(d, m) be given by
b(s, !, ⇣) = a(s , !, ⇣).
(7.3.5)
Lemma 7.15. Suppose the functional a satisfies (7.3.2)-(7.3.4).
(i) For all t
0,
(!, ⇣) 7! a(t, !, ⇣) is Ft ⌦ B(Cd ) measurable.
(7.3.6)
(ii) For any continuous (F⇧ ) adapted process V , Z defined by Zt = a(t, ·, V ) ( i.e. Zt (!) =
a(t, !, V (!)) ) is an r.c.l.l. (F⇧ ) adapted process.
7.3. Stochastic Di↵erential Equations
225
(iii) For any stop time ⌧ ,
(!, ⇣) 7! a(⌧ (!), !, ⇣) is F⌧ ⌦ B(Cd ) measurable.
(7.3.7)
T
Proof. Since for fixed t, ⇣, the mapping ! 7! a(t, !, ⇣) is Ft measurable and in view of
(7.3.4), ⇣ 7! a(t, !, ⇣) is continuous for fixed t, !, it follows that (7.3.6) is true since Cd is
separable.
For part (ii), let us define a process V t by Vst = Vs^t . In view of assumption (7.3.3), Z
is an r.c.l.l. process.The fact that ! 7! V t (!) is Ft measurable along with (7.3.6) implies
that Zt = a(t, ·, V t ) is Ft measurable.
For part (iii), when ⌧ is a simple stop time, (7.3.7) follows from (7.3.6). For a general
bounded stop time ⌧ , the conclusion (7.3.7) follows by approximating ⌧ from above by
simple stop times and using right continuity of a(t, !, ⇣). For a general stop time ⌧ , (7.3.7)
follows by approximating ⌧ by ⌧ ^ n.
F
Let b(t, !, ⇣) = a(t , !, ⇣). Let 0 denote the process that is identically equal to zero.
Since (t, !) 7! b(t, !, 0) is an r.c.l.l. adapted process, using hypothesis (7.3.4), it follows
that for ⇣ 2 Cd
sup ka(t, !, ⇣)k  Kt0 (!)(1 + sup |⇣(s)|)
(7.3.8)
0st
A
where
0st
Kt0 (!) = Kt (!) + sup ka(t, !, 0)k.
(7.3.9)
0st
D
R
Kt0 is clearly an r.c.l.l. adapted process.
Here too, as in the Brownian motion case, a continuous (Rd -valued) adapted process U
is said to be a solution to the equation (7.3.1) if
Z t
U t = ⇠0 +
b(s, ·, U )dYs
(7.3.10)
i.e. for 1  j  d,
0+
Utj
=
⇠0j
+
m Z
X
k=1
t
0+
bjk (s, ·, U )dYsk
where U = (U 1 , . . . , U d ) and b = (bjk ).
It is convenient to introduce matrix and vector-valued processes and stochastic integral
R
f dX where f is matrix-valued and X is vector-valued. All our vectors are column vectors,
though we will write it as c = (c1 , c2 , . . . , cm ).
Let X 1 , X 2 , . . . X m be continuous semimartingales w.r.t. the filtration (F⇧ ). We will
say that X = (X 1 , X 2 , . . . X m ) is an Rd -valued semimartingale. Similarly, for 1  j  d,
226
Chapter 7. Continuous Semimartingales
1  k  m let fjk be locally bounded predictable process. f = (fjk ) will be called an
R
L(d, m)-valued locally bounded predictable process. The stochastic integral Y = f dX is
defined as follows: Y = (Y 1 , Y 2 , . . . , Y d ) where
m Z
X
j
Y =
fjk dX k .
k=1
Let us recast the growth estimate in matrix-vector form:
Theorem 7.16. Let X = (X 1 , X 2 , . . . X m ), where X j is a amenable semimartingale for
each j, 1  j  m. Then for any locally bounded L(d, m)-valued predictable f , and a stop
time , one has
Z s
Z ^T
2
E[ sup |
f dX| ]  2m(4 + T )E[
kfs k2 ds ].
(7.3.11)
E[ sup |
0s ^T
Z
0+
s
T
Proof.
0+
f dX|2 ]
0+
j=1
E[ sup |
m Z
X
0s ^T k=1
s
0+
A

d
X
F
0s ^T
j=1 k=1
R
m
d X
m
X
E[ sup |
 m(8 + 2T )
0s ^T
d X
m
X
j=1 k=1
Z ^T
D
= 2m(4 + T )E[
0+
E[
Z
Z
fjk dX k |2 ]
s
0+
fjk dX k |2 ]
^T
0+
|fjk (s)|2 ds]
kfs k2 ds].
In the second inequality above we have used the estimate (7.2.3).
We will prove existence and uniqueness of solution of (7.3.1). When the driving semimartingale satisfies (7.2.2), the proof is almost the same as the proof when the driving
semimartingale is a Brownian Motion. We will prove this result without making any integrability assumptions on the initial condition ⇠0 and the uniqueness assertion is without
any moment condition. For this the following simple observation is important.
Remark 7.17. Suppose M is a square integrable martingale w.r.t. a filtration (F⇧ ) and let
⌦0 2 F0 . Then Nt = 1⌦0 Mt is also a square integrable martingale and further [N, N ]t =
7.3. Stochastic Di↵erential Equations
227
1⌦0 [M, M ]t . Thus, the estimate (5.3.22) can be recast as
E[1⌦0 sup |Ms |2 ]  4E[ 1⌦0 [M, M ]T ].
0sT
Using this we can refine the estimate (7.2.3): For a amenable semimartingale X, any locally
bounded predictable f , a stop time and ⌦0 2 F0 , one has
Z s
Z ^T
2
f dX| ]  2(4 + T )E[1⌦0
|fs |2 ds].
(7.3.12)
E[1⌦0 sup |
0s ^T
0+
0+
here is the modified estimate for vector-valued case, a modification of (7.3.11): if X =
(X 1 , X 2 , . . . X m ) where each X j is a amenable semimartingale and f = (fjk ) is an L(d, m)valued locally bounded predictable process, then
Z s
Z ^T
2
E[1⌦0 sup |
f dX| ]  2m(4 + T )E[1⌦0
kfs k2 ds].
(7.3.13)
0+
0+
T
0s ^T
We will first prove uniqueness of solution in the special case when the driving semimartingale is a amenable semimartingale.
A
F
Theorem 7.18. Let Y = (Y 1 , Y 2 , . . . Y m ) where Y j is a amenable continuous semimartingale for each j. Let the functional a satisfy conditions (7.3.2)- (7.3.4) and b be defined by
(7.3.5). Let ⇠0 be any F0 measurable random variable. Then if U, Ũ are (F⇧ ) adapted
continuous process satisfying
Z t
U t = ⇠0 +
b(s, ·, U )dYs ,
(7.3.14)
R
Ũt = ⇠0 +
D
then
Z
0+
t
0+
b(s, ·, Ũ )dYs .
P(Ut = Ũt 8t
(7.3.15)
0) = 1.
(7.3.16)
Proof. For i
1, let ⌧i = inf{t
0 : Kt0 (!)
i or Kt0 (!)
i} ^ i where Kt0 (!) is
the r.c.l.l. adapted process given by (7.3.9). Thus each ⌧i is a stop time, ⌧i " 1 and for
0  t < ⌧i (!), we have
0  Kt (!)  Kt0 (!)  i.
Recalling that b(t, ·, ⇣) = a(t , ·, ⇣), we conclude that for ⇣, ⇣1 , ⇣2 2 Cd
sup
0s(t^⌧i (!))
kb(s, !, ⇣2 )
b(s, !, ⇣1 )k  i
sup
0s(t^⌧i (!))
|⇣2 (s)
⇣1 (s)|.
(7.3.17)
and
sup
0s(t^⌧i (!))
kb(s, !, ⇣)k  i(1 +
sup
0s(t^⌧i (!))
|⇣(s)|).
(7.3.18)
228
Chapter 7. Continuous Semimartingales
We first show that if V is any solution to (7.3.14), i.e. V satisfies
Z t
Vt = ⇠ 0 +
b(s, ·, V )dYs
(7.3.19)
0+
then for k
1, i
1,
E[1{|⇠0 |k} sup |Vt |2 ] < 1.
(7.3.20)
0t⌧i
Let us fix k, i for now. For j 1, let j = inf{t 0 : |Vt | j}. Since V is a continuous
adapted process with V0 = ⇠0 , it follows that j is a stop time, limj!1 j = 1 and
sup
0t(u^⌧i ^
j)
|Vt |2  max(|⇠0 |2 , j 2 ).
(7.3.21)
Thus using the estimate (7.3.12) along with (7.3.18), we get for, i, j, k
sup
0t(u^⌧i ^
j)
|Vt |2 ]
2
 2[E[1{|⇠0 |k} (|⇠0 | + 2m(4 + i)
Z
(u^⌧i ^
0+
(u^⌧i ^
j)
kb(s, ·, Vs )k2 ds)]]
j)
F
Z
T
E[1{|⇠0 |k}
Writing
j (u)
j (u)
sup
0t(u^⌧i ^
R
it follows that for 0  u  T ,
= E[1{|⇠0 |k}
(1 +
0+
A
 E[1{|⇠0 |k} (2k 2 + 8m(4 + i)i2
2
j)
sup
0t(s^⌧i ^
3
2
Z
u
j (s)ds]
0+
is a bounded function. Thus, using (Gronwall’s) Lemma
D
j
j)
|Vs |2 )ds)].
|Vt |2 ],
 [2E[1{|⇠0 |k} |⇠0 | ] + 8m(4 + i)i + 8m(4 + i)i
and further, (7.3.21) yields that
3.23, we conclude
1
j (u)
 [8m(4 + i)i3 + 2k 2 ] exp{8m(4 + i)i2 u}, 0  u  i.
Now letting j increase to 1 we conclude that (7.3.20) is true.
Returning to the proof of (7.3.16), since U, Ũ both satisfy (7.3.15), both also satisfy
(7.3.20) and hence we conclude
E[1{|⇠0 |k} sup |Ut
0t⌧i
Now
Ut
Ũt =
Z
Ũt |2 ] < 1.
t
0+
(b(s, ·, U )
b(s, ·, Ũ ))dYs
(7.3.22)
7.3. Stochastic Di↵erential Equations
229
and hence using the Lipschitz condition (7.3.17) and the growth estimate (7.3.13), we
conclude that for 0  t  T ,
Z (t^⌧i )
E[1{|⇠0 |k} sup |Us Ũs |2 ]  2m(4 + i)i2 E[
1{|⇠0 |k} sup |Us Ũs |2 ]du
and hence the function
0su
0+
0s(t^⌧i )
defined by
(t) = E[1{|⇠0 |k}
satisfies, for a suitable constant Ci
(t)  Ci
Z
sup
0s(t^⌧i )
Ũs |2 ]
|Us
t
(u)du, t
0.
0+
P({|⇠0 |  k} \ {
1 and k
|Us
Ũs |
✏}) = 0
1. Since ⌧i " 1, this proves (7.3.16) completing the proof.
F
for all ✏ > 0, i
sup
0s(t^⌧i )
T
As noted above (see (7.3.22)) (t) < 1 for all t. Now (Gronwall’s) Lemma 3.23 implies
that (t) = 0 for all t. Thus we conclude
D
R
A
We have thus seen that if the Y is a amenable semimartingale then uniqueness of
solution holds for the SDE (7.3.14) and the proof is on the lines of the proof in the case
of Brownian motion. One big di↵erence is that uniqueness is proven without a priori
requiring the solution to satisfy a moment condition. This is important as while the
stochastic integral is invariant under time change, moment conditions are not. And this
is what enables us to prove uniqueness when the driving semimartingale may not be a
amenable semimartingale.
Using random time change we extend the result on uniqueness of solutions to the SDE
(7.3.14) to the case when the driving semimartingale may not be a amenable semimartingale.
Theorem 7.19. Let X = (X 1 , X 2 , . . . X m ) where each X j is a continuous semimartingale.
Let the functional a satisfy conditions (7.3.2) - (7.3.4) and b be defined by (7.3.5). Let ⇠0 be
any F0 measurable random variable. If V, Ṽ are (F⇧ ) adapted continuous process satisfying
Z t
Vt = ⇠0 +
b(s, ·, V )dXs ,
(7.3.23)
Ṽt = ⇠0 +
Then
Z
0+
t
0+
b(s, ·, Ṽ )dXs .
P(Vt = Ṽt 8t
0) = 1.
(7.3.24)
(7.3.25)
230
Chapter 7. Continuous Semimartingales
Proof. Let
be a (F⇧ ) random time change such that Y j = [X j ], 1  j  m are
amenable semimartingales (such a random time change exists as seen in Theorem 7.14).
Let (G⇧ ) = ( F⇧ ), = [ ] 1 be defined via (7.1.5).
We define c(t, !, ⇣), d(t, !, ⇣) as follows: Fix ! and let ✓! (⇣) be defined by
✓! (⇣) = ⇣(
s (!))
(7.3.26)
and let c(t, !, ⇣) = a( t (!), !, ✓! (⇣)), d(t, !, ⇣) = b( t (!), !, ✓! (⇣)). Since
d(t, !, ⇣) = d(t , !, ⇣).
We will first observe that for all ⇣1 , ⇣2 2 Cd ,
sup kc(u, !, ⇣2 )
is continuous,
c(u, !, ⇣1 )k
= sup ka(
u (!), !, ✓! (⇣1 ))
 K s (!)
sup
0v
 K s (!)
0v
s (!)
sup
s (!)
|✓! (⇣1 )(v)
|⇣2 (
 K s (!) sup |⇣2 (u)
0us
a(
u (!), !, ✓! (⇣2 ))k
✓! (⇣2 )(v)|
v (!))
⇣1 (
(7.3.27)
v (!))|
F
0us
T
0us
⇣1 (u)|.
A
We now prove that for each ⇣,
(t, !) 7! c(t, !, ⇣) is an r.c.l.l. (G⇧ ) adapted process.
(7.3.28)
R
That the mapping is r.c.l.l. follows since a is r.c.l.l. and t is continuous strictly increasing
function. To see that it is adapted, fix t and let ⇣ t be defined by ⇣ t (s) = ⇣(s ^ t). In view
of (7.3.27), it follows that
Now
D
c(t, !, ⇣) = c(t, !, ⇣ t ) = a( t (!), !, ✓! (⇣ t )).
✓! (⇣ t ) = ⇣ t (
s (!))
= ⇣(
s (!)
(7.3.29)
^ t).
Since s is a (G⇧ )-stop time, it follows that ! 7! ✓! (⇣ t ) is Gt = F t measurable. Since t
is a (F⇧ )-stop time, part (iii) in Lemma 7.15 gives that ! 7! a( t (!), !, ⇣) is F t ⌦ B(Cd )
measurable. From this we get
! 7! a( t (!), !, ✓! (⇣ t )) is Gt measurable.
(7.3.30)
The relation (7.3.28) follows from (7.3.29) and (7.3.30). Let H = [K]. Then H is an (G⇧ )
adapted increasing process and (7.3.27) can be rewritten as
sup kc(u, !, ⇣2 )
0as
c(u, !, ⇣1 )k  Hs (!) sup |⇣2 (u)
0us
⇣1 (u)|.
(7.3.31)
7.3. Stochastic Di↵erential Equations
231
Let U = [V ], Ũ = [Ṽ ]. Then recall V =
As = b(s, ·, V ) and Ãs = b(s, ·, Ṽ ). Then
Bs = A
= b(
[U ], Ṽ =
[Ũ ]. Let A, Ã be defined by
s
s , ·,
[U ])
= d(s, ·, U )
and likewise B̃t = d(t, Ũ ).
Thus the processes U, Ũ satisfy
Ũt = ⇠ +
Z
t
d(s, U )dỸs ,
0+
t
d(s, Ũ )dỸs .
0+
Since c, d satisfy (7.3.2)-(7.3.5), Theorem 7.18 implies
P(Ut = Ũt 8t
T
Ut = ⇠ +
Z
F
0) = 1
which in turn also proves (7.3.25) since V = [U ] and Ṽ = [Ũ ].
A
We are now ready to prove the main result on existence of solution to an SDE driven by
continuous semimartingales. Our existence result is a modification of Picard’s successive
approximation method. Here the approximations are explicitly constructed.
D
R
Theorem 7.20. Let X 1 , X 2 , . . . X m be continuous semimartingales. Let the functional a
satisfy conditions (7.3.2)-(7.3.4) and b be defined by (7.3.5). Let ⇠ be any F0 measurable
random variable. Then there exists a (F⇧ ) adapted continuous process V that satisfies
Z t
Vt = ⇠ 0 +
b(s, ·, V )dXs .
(7.3.32)
0+
Proof. We will construct approximations V (n) that converge to a solution of (7.3.32).
(0)
Let Vt
= ⇠0 . The processes V (n) are defined by induction on n. Assuming that
V (0) , . . . , V (n 1) have been defined, we now define V (n) : Fix n.
(n)
Let ⌧0
(n)
⌧j+1
= 1 and if
(n)
(n)
= 0 and let {⌧j
(n)
⌧j
: j
(n)
1} be defined inductively as follows: if ⌧j
< 1 then
(n)
⌧j+1 = inf{s > ⌧j
: ka(s, ·, V (n
1)
)
or ka(s , ·, V (n
(n)
a(⌧j , ·, V (n
1)
)
(n)
1)
)k
a(⌧j , ·, V (n
2
1)
)k
= 1 then
n
2
n
}.
(7.3.33)
232
Chapter 7. Continuous Semimartingales
Since the process s 7! a(s, ·, V (n
(n)
is a stop time and limj"1 ⌧j
(n)
Vt
(n)
1) )
is an adapted r.c.l.l. process, it follows that each ⌧j
(n)
= 1. Let V0
=V
(n)
(n)
⌧j
(n)
= ⇠0 and for j
(n)
+ a(⌧j , ·, V (n
1)
)(Xt
0, ⌧j
(n)
< t  ⌧j+1 let
X⌧ (n) ).
j
Equivalently,
(n)
Vt
= ⇠0 +
1
X
j=0
(n)
a(⌧j , ·, V (n
1)
)(Xt^⌧ (n)
j+1
Xt^⌧ (n) ).
(7.3.34)
j
Thus we have defined V (n) and we will show that these process converge almost surely and
the limit process V is the required solution. Let F (n) , Z (n) , R(n) be defined by
(n)
Zt
=b(t, ·, V (n 1) )
Z t
=⇠0 +
Fs(n) dXs
0+
=
1
X
j=0
(n)
(n)
a(⌧j , ·, V (n
=⇠0 +
Z
t
0+
1)
)1
(7.3.36)
(⌧
(n) (n)
,⌧
]
j
j+1
(t)
(7.3.37)
F
(n)
Rt
Vt
(7.3.35)
T
(n)
Ft
Rs(n) dXs .
A
(n)
Let us note that by the definition of {⌧j
(n)
: j
(n)
|Ft
(7.3.38)
0}, we have
Rt |  2
n
.
(7.3.39)
R
We will prove convergence of V (n) employing the technique used in the proof of uniqueness to the SDE Theorem 7.19- namely random time change.
D
Let be a (F⇧ ) random time change such that Y j = [X j ], 1  j  m are amenable
semimartingales (such a random time change exists as seen in Theorem 7.14). Let (G⇧ ) =
( F⇧ ), = [ ] 1 be defined via (7.1.5).
Let c(t, !, ⇣), d(t, !, ⇣) be given by (7.3.26). As noted in the proof of Theorem 7.19, c, d
(n)
satisfy (7.3.2)-(7.3.5). We will transform {⌧i : n 1, i 1}, {V (n) , F (n) , Z (n) , R(n) :
n 1} to the new new time scale. Thus for n 1, j 0 let
(n)
j
(n)
= [⌧j ], U (n) = [V (n) ], G(n) = [F (n) ], W (n) = [Z (n) ], S (n) = [R(n) ].
Now it can be checked that
(n)
j+1
= inf{s >
(n)
j
: ka(s, ·, U (n
1)
)
or ka(s , ·, U (n
a(
1)
)
(n)
(n 1)
)k 2 n
j , ·, U
(n)
a( j , ·, U (n 1) )k
2
n
}.
(7.3.40)
7.3. Stochastic Di↵erential Equations
(n)
j
(n)
j
is a stop time and limj"1
(n)
Ut
=⇠0 +
1
X
(n)
(n 1)
)(Yt^
j , ·, U
c(
j=0
(n)
Gt
(n)
Wt
(n)
= 1. Further, U0
= ⇠0 and
Yt^
(n)
j+1
(n)
j
),
(7.3.41)
=d(t, ·, U (n 1) ),
Z t
=⇠0 +
G(n)
r dYr ,
(7.3.42)
(7.3.43)
0+
(n)
St
=
1
X
j=0
(n)
Ut
(n)
(n 1)
)1
j , ·, U
(
c(
=⇠0 +
Also, we have
Z
t
(n) (n)
,
]
j
j+1
(t),
(7.3.44)
Sr(n) dYr .
0+
(n)
(7.3.45)
(n)
|Gt
St |  2
T
Each
233
n
.
(7.3.46)
0 : |Ht |
⇢j = inf{t
Then ⇢j are (G⇧ )-stop times and
sup
0s(t^⇢j )
|Ws(n)
c(a, !, ⇣1 )k  j
Ws(n
sup
0a(s^⇢j )
|⇣2 (a)
Z
(t^⇢j )
sup
0
2
Likewise, using (7.3.46) we get for n
0s(t^⇢j )
|Ws(n)
Z
(7.3.47)
2 (using (7.3.13) along with
0s(r^⇢j )
|G(n)
s
(t^⇢j )
sup
0
0s(r^⇢j )
|Us(n
G(n
s
1)
1) 2
| dr]
Us(n
(7.3.48)
2) 2
| dr]
1,
Us(n) |2 ]
 2m(4 + j)E[1{|⇠0 |k}
 2m(4 + j)4
⇣1 (a)|.
| ]
 2m(4 + j)j E[1{|⇠0 |k}
sup
j} ^ j.
1) 2
 2m(4 + j)E[1{|⇠0 |k}
E[1{|⇠0 |k}
⇠0 |
" 1. Further, for all ⇣1 , ⇣2 2 Cd we have
1 (fixed), we observe that for all n
D
E[1{|⇠0 |k}
j or |Ut
A
kc(a, !, ⇣2 )
Now for k 1 and j
(7.3.42) and (7.3.43))
j
(1)
j or |Ht |
R
sup
0a(s^⇢j )
F
Recall that we had shown in (7.3.31) that c satisfies Lipschitz condition with coefficient
H = [K]. For j 1, let
n
j.
Z
(t^⇢j )
sup
0
0s(r^⇢j )
|G(n)
s
Ss(n) |2 dr]
(7.3.49)
234
Chapter 7. Continuous Semimartingales
Combining (7.3.48) and (7.3.49), we observe that for n
x, y, z, (x + y + z)2  3(x2 + y 2 + z 2 ))
E[1{|⇠0 |k}
sup
0s(t^⇢j )
|Us(n)
Us(n
+ 6m(4 + j)j 2 E[1{|⇠0 |k}
Z
2 (using for positive numbers
1) 2
n
| ]  6m(4 + j)(4
)j
(7.3.50)
(t^⇢j )
sup
0
(n 1)
+4
0s(r^⇢j )
|Us(n
1)
Us(n
Us(n
1) 2
2) 2
| dr]
Let
f (n) (t) = E[1{|⇠0 |k}
sup
0s(t^⇢j )
|Us(n)
| ]
(7.3.51)
0
(0)
Ut
Since
= ⇠0 , by the definition of ⇢j ,
induction on n) that
(1)
ft
T
Then, writing Cm,j = 30m(4 + j)j 2 , (7.3.50) implies for n 2
Z t
f (n) (t) = Cm,j + Cm,j
f (n 1) (r)dr.
 j 2  Cm,j . Now (7.3.51) implies (via
(Cm,j t)n
n!
(7.3.52)
F
f (n) (t)  Cm,j .
1
X
n=1
k1{|⇠0 |k} sup |Us(n)
0s⇢j
1
X
R
and as a consequence
A
and as a consequence (writing |·|2 for the L2 (P ) norm, and recalling ⇢j  j)
k1{|⇠0 |k}
sup |Us(n)
n=1 0s⇢j
Us(n
1)
Us(n
1)
|k2 < 1.
|k2 < 1.
D
As in the proof of Theorem 3.26, it now follows that for k
Nk,j = {! : 1{|⇠0 (!)|k}
Since for all T < 1
sup
(
1
X
0s⇢j (!) n=1
|Us(n) (!)
1, P(Nk,j ) = 0 where
1, j
Us(n
(7.3.53)
1)
(!)|) < 1}.
P([1
k,j=1 {! : |⇠0 (!)|  k, ⇢j  T }) = 1
(n)
it follows that P(N ) = 0 where N = [1
k,j=1 Nk,j and for ! 62 N , Us (!) converges uniformly
on [0, T ] for every T < 1. So let us define U as follows:
8
(n)
<lim
c
n!1 Ut (!) if ! 2 N
Ut (!) =
:0
if ! 2 N.
7.3. Stochastic Di↵erential Equations
235
By definition, U is a continuous (G⇧ ) adapted process (since by assumption N 2 F0 = G0 )
and U (n) converges to U uniformly in [0, T ] for every T almost surely (and thus also
ducp (U (n) , U ) ! 0). Also, (7.3.53) yields
lim k1{|⇠0 |k} sup |Us(n)
n,r!1
Us(r) |k2 = 0.
0s⇢j
Now Fatou’s Lemma implies
lim k1{|⇠0 |k} sup |Us(n)
n!1
Us |k2 = 0
0s⇢j
which is same as
lim E[1{|⇠0 |k} sup |Us(n)
0s⇢j
Now defining Gt = d(t, ·, U ) and Wt = ⇠0 +
(7.3.47) along with (7.3.54) that
Rt
0
(7.3.54)
Gs dYs , it follows using the Lipschitz condition
lim E[1{|⇠0 |k} sup |Ws(n)
n!1
Us |2 ] = 0.
0s⇢j
T
n!1
Ws |2 ] = 0.
1)
F
Now (7.3.49), (7.3.54) and (7.3.55) imply that (for all j, k
(7.3.55)
E[1{|⇠0 |k} sup |Ws
0s⇢j
Us |2 ] = 0
A
and hence that
P(Ws = Us 8s
0) = 1.
R
Recalling the definition of G, W , it follows that U satisfies
Z t
U t = ⇠0 +
d(s, ·, U )dYs .
0
It now follows that V = [U ] satisfies (7.3.32).
D
We have shown the existence and uniqueness of solution to the SDE (7.3.32). Indeed,
we have explicitly constructed processes V (n) that converge to V . We record this in the
next theorem
Theorem 7.21. Let X 1 , X 2 , . . . X m be continuous semimartingales. Let a, b satisfy conditions (7.3.2)-(7.3.5). Let ⇠ be any F0 measurable random variable. For n
1 let
(n)
(n)
{⌧j : j
1} and V
be defined inductively by (7.3.33) and (7.3.34) as in the proof
of Theorem 7.20. Let V be the (unique) solution to the SDE
Z t
Vt = ⇠ 0 +
b(s, ·, V )dXs .
0+
Then V
(n) em
!
V and
(n)
Vt
converges to Vt in ucc topology almost surely.
236
Chapter 7. Continuous Semimartingales
Proof. We had constructed in the proof of Theorem 7.20 a (F⇧ ) random time change
and filtration (G⇧ ) = ( F⇧ ) such that U (n) = [V (n) ] converges to U in ucp metric:
1 is also a (G ) random time change, V (n) = [U (n) ] and
ducp (U (n) , U ) ! 0. Now =
⇧
V = [U ]. It follows from Theorem 7.6 that ducp (V (n) , V ) ! 0. Let Ft = b(t, ·, V ). Now
ucp
the Lipschitz condition on a, b- (7.3.4) implies F (n) ! F , where F (n) is defined by (7.3.35)
Rt
ucp
and then (7.3.39) implies R(n) ! F . Now Vt = ⇠0 + 0 Fs dXs and (7.3.38) together imply
em
that V (n) ! V . As for almost sure convergence in ucc topology, we had observed that it
holds for U (n) , U and then we can see that same holds for V (n) , V
7.4
Pathwise Formula for solution of SDE
T
In this section, we will consider the SDE
dVt = f (t , H, V )dXt
(7.4.1)
F
where f : [0, 1) ⇥ Dr ⇥ Cd 7! L(d, m), H is an Rr -valued r.c.l.l. adapted process and X
is a continuous semimartingale. Here Dr = D([0, 1), Rd ), Cd = C([0, 1), Rd ). For t < 1,
⇣ 2 Cd and ↵ 2 Dr , let ↵t (s) = ↵(t ^ s) and ⇣ t (s) = ⇣(t ^ s). We assume that f satisfies
(7.4.2)
t 7! f (t, ↵, ⇣) is an r.c.l.l. function 8↵ 2 Dr , ⇣ 2 Cd
(7.4.3)
A
f (t, ↵, ⇣) = f (t, ↵t , ⇣ t ), 8↵ 2 Dr , ⇣ 2 Cd , 0  t < 1,
kf (t, ↵, ⇣1 )
R
and that there exists constants for T < 1, CT < 1 such that 8↵ 2 Dr , ⇣1 , ⇣2 2 Cd , 0 
tT
f (t, ↵, ⇣2 )k  CT (1 + sup |↵(s)|)( sup |⇣1 (s)
0st
⇣2 (s)|).
(7.4.4)
that yields a pathwise solution to the
D
As in section 6.2, we will now obtain a mapping
SDE (7.4.1).
0st
Theorem 7.22. Suppose f satisfies (7.4.2), (7.4.3) and (7.4.4). Then there exists a mapping
: Rd ⇥ Dr ⇥ Cd 7! C([0, 1), L(d, m))
such that for an adapted r.c.l.l. process H and a continuous semimartingale X,
V =
yields the unique solution to the SDE
Vt = ⇠ 0 +
Z
(⇠0 , H, X)
t
f (s , H, V )dX.
0
(7.4.5)
7.4. Pathwise Formula for solution of SDE
237
Proof. We will define mappings
(n)
: Rd ⇥ Dr ⇥ Cd 7! C([0, 1), L(d, m))
0. Let (0) (u, ↵, ⇣)(s) = u for all s 0. Having defined
(n) as follows. Fix n and u 2 Rd , ↵ 2 D and ⇣ 2 C .
r
d
inductively for n
(n 1) , we define
(n)
(n)
(0) ,
(1) , . . . ,
(n)
Let t0 = 0 and let {tj : j 1} be defined inductively as follows: ({tj : j 1} are
themselves functions of (u, ↵, ⇣), which are fixed for now and we will suppress writing it as
(n)
(n)
(n)
a function) if tj = 1 then tj+1 = 1 and if tj < 1 then writing
(n 1)
(n 1)
(u, ↵, ⇣)(s) = f (s, ↵,
(u, ↵, ⇣)),
let
:k
(n 1)
or k
(since
(n 1) (u, ↵, ⇣)
(n)
(n 1)
(u, ↵, ⇣)(s )
(n)
is an r.c.l.l. function, tj
(u, ↵, ⇣)(s) = u +
1
X
(n 1)
j=0
(n) (u, ↵, ⇣).
(n 1)
n
2
(n)
(u, ↵, ⇣)(tj )k
2
n
}
" 1 as j " 1) and
(n)
(u, ↵, ⇣)(s)(⇣(t ^ tj+1 )
(n)
⇣(t ^ tj )).
Now we define
n
(n) (u, ↵, ⇣)
:0
if the limit exists in ucc topology
(7.4.6)
otherwise.
R
(u, ↵, ⇣) =
8
<lim
(n)
(u, ↵, ⇣)(tj )k
A
This defines
(n 1)
(u, ↵, ⇣)(s)
T
(n)
tj
F
(n)
t